Tarasov - Basic Concepts of Quantum Mechanics - 2021
Tarasov - Basic Concepts of Quantum Mechanics - 2021
Tarasov - Basic Concepts of Quantum Mechanics - 2021
Concepts
limited to microphenomena.
In our endless quest for
understanding and perfecting
antum
Mechanics
BASIC CONCEPTS OF
Q UA N T U M M E C H A N I C S
L.V. Tarasov
Basic Concepts of Quantum Mechanics
Preface 9
3 Uncertainty Relations 38
10 Superposition of States 95
Interlude: Are These the Same Waves?, Or, Again on Waves in Quantum Mechanics
112
Appendices 243
Bibliography 259
Preface
Research in physics, conducted at the end of the 19th century and in the Some Preliminary Remarks
th
first half of the 20 century, revealed exceptionally peculiar nature of the
laws governing the behaviour of microparticles-atoms, electrons, and so
on. On the basis of this research a new physical theory called quantum
mechanics was founded.
It is also not surprising that even today anyone who starts studying
quantum mechanics encounters some sort of psychological barrier. This
10
The aim of this book is to acquaint the reader with the concepts and The Structure of the Book
ideas of quantum mechanics and the physical properties of matter; to
reveal the logic of its new ideas, to show how these ideas are embodied
in the mathematical apparatus of linear operators and to demonstrate the
working of this apparatus using a number of examples and problems of
interest to engineering students.
The second chapter deals with the physical concepts of quantum me-
chanics. The chapter starts with an analysis of a set of basic experiments
12
Finally, the book contains many quotations. The author is sure that
the “original words” of the founders of quantum mechanics will offer the
reader useful additional information.
The author wishes to express his deep gratitude to Prof. I.I. Gurevich, Personal Remarks
Corresponding Member of the USSR Academy of Sciences, for the stim-
ulating discussions which formed the basis of this book. Prof. Gurevich
13
discussed the plan of the book and its preliminary drafts, and was kind
enough to go through tho manuscript. His advice not only helped mould
the structure of the book, but also helped in the nature of exposition of
the material. The subsection “The Essence of Quantum Mechanics” in
Section 16 is a direct consequence of Prof. Gurevich’s ideas.
The author would like to record the deep impression left on him by
the works on quantum mechanics by the leading American physicist R.
Feynman.3 While reading the sections in this book dealing with the ap- 3 Feynman, R. P. and Hibbs, A. (1965).
Quantum mechanics and path integrals.
plications of the idea of probability amplitude, superposition principle, McGraw-Hill, N.Y; and Feynman, R. P.,
microparticles with two basic states, the reader can easily detect a definite Leighton, R. B., and Sands, M. (1965).
Lectures on Physics, volume 3. Addison-
similarity in approach with the corresponding parts in Feynman’s “Lec- Wesley Reading, Mass
tures in Physics”. The author was also considerably influenced by N. Bohr
(in particular by his wonderful essays Atomic Physics and Human Knowl-
edge)4 , V. A. Fock5 , W. Pauli6 , P. Dirac7 , and also by the comprehensive 4 Bohr, N. (1958a). Atomic Physics and
Human Knowledge, volume 21. New York,
works of L. D. Landau and E. M. Lifshitz8 , D. I. Blokhintsev9 , E. Fermi10 , L. Wiley
Schiff11 . 5 Fock, V. A. (1978). Fundamentals
of Quantum Mechanics. Mir Publishers,
Moscow; and Fock, V. A. (1957). On the
The author is especially indebted to Prof. M. I. Podgoretsky, D.Sc., for interpretation of quantum mechanics.
a thorough and extremely useful analysis of the manuscript. He is also Czechoslovak Journal of Physics, 7:643–656
6 Pauli, W. (1946). Die allgemeinen
grateful to Prof. Yu. A. Vdovin, Prof. E. E. Lovetsky, Prof. G. F. Drukarev, prinzipien der wellenmechanik. In Geiger,
Prof. V. A. Dyakov, Prof. Yu. N. Pchelnikov, and Dr. A. M. Polyakov, all H. and Scheel, K., editors, Handbuch der
Physik, volume 24, pages 83–272. J.W.
of whom took the trouble of going through the manuscript and made a Edwards, 2nd edition edition
7 Dirac, P. A. M. (1958). The Principles
number of valuable comments. Lastly, the author is indebted to his wife
of Quantum Mechanics. Clarendon Press
Aldina Tarasova for her constant interest in the writing of the book and Oxford
8 Landau, L. D. and Lifshitz, E. M.
her help in the preparation of the manuscript. But for her efforts, it would
(1977). Quantum Mechanics (Non-
have been impossible to bring the book to its present form. relativistic Theory). Pergamon
9 Blokhintsev, D. I. (1964). Principles of
Quantum Mechanics. D. Reidel Publishing
Co
10 Fermi, E. (1961). Notes on quantum me-
chanics. University of Chicago Press
11 Schiff, L. I. (1968). Quantum mechanics.
McGraw-Hill, N.Y., 3rd edition
Prelude: Can the System of
aut hor: It is well known that the basic contents of a physical the- Participants: the Author
ory are formed by a system of concepts which reflect the objective and the Classical Physicist
laws of nature within the framework of the given theory. Let us (Physicist of the older gener-
take the system of concepts lying at the root of classical physics. ation, whose views have been
Can this system be considered logically perfect? formed on the basis of classical
aut hor: It means that classical physics reduces the question “what
is an object like?” to “what is it made of?”
aut hor: But surely such a step will destroy the notion of the object
or phenomenon as a single unit.
aut hor: I would like to add that one special consequence of the
“principle of analysis” is the notion, characteristic of classical
physics, of the mutual independence of the object of observation
and the measuring instrument (or observer). We have an instrument
and an object of measurement. They can and should be considered
separately, independently from one another.
aut hor: And yet there are grounds to doubt the “flawlessness” of
classical concepts even from very general considerations.
Molecules, atoms, atomic nuclei and elementary particles belong to the Microparticles
category of microparticles. The list of elementary particles is at present
fairly extensive and includes quanta of electromagnetic field (photons) as
well as two groups of particles, the hadrons and the leptons. Hadrons are
characterized by a strong (nuclear) interaction, while leptons never take
part in strong interactions. The electron, the muon and the two neutrinos
(the electronic and muonic) are leptons. The group of hadrons is numer-
ically much larger. It includes nucleons (proton and neutron), mesons
(a group of particles lighter than the proton) and hyperons (a group of
particles heavier than the neutron). With the exception of photons and
some neutral mesons, all elementary particles have corresponding anti-
particles.
Spin is one of the most important specific characteristics of a micropar- Spin of a Microparticle
ticle. It may be interpreted as the angular momentum of the microparticle
not related to its motion as a whole (it is frequently known as the inter-
nal angular momentum of the microparticle). The square of this angular
momentum is equal to ℏ2𝑠 (𝑠 + 1), where 𝑠 for the given microparticle is
a definite integral or semi-integral number (it is this number which is
usually referred to as the spin), ℏ is a universal physical constant which
plays an exceptionally important role in quantum mechanics. It is called
24 bas i c co n c e p ts o f q ua n t u m m e c h a ni cs
If we know the spin of a microparticle, we can predict its behaviour Bosons and Fermions
in the collective of microparticles similar to it (in other words, to predict
the statistical properties of the microparticle). It turns out that all the
microparticles in nature can be divided into two groups, according to their
statistical properties: a group with integral values of spin or with zero
spin, and another with half-integral spin.
All elementary particles except the photon, the electron, the proton Instability of Microparticles
and both neutrinos are unstable. This means that they decay sponta-
p h y s i c s o f t h e m i c r o pa r t i c l e s 25
neously, without any external influence, and are transformed into other
particles. For example, a neutron spontaneously decays into a proton, an
electron and an electronic antineutrino:
𝑛 → 𝑝 + 𝑒 − + 𝜈𝑒
𝜋 + → 𝜇+ + 𝜈𝜇,
𝜋 + → 𝑒 + + 𝜈𝑒 ,
𝜋 + → 𝜋 0 + 𝑒 + + 𝜈𝑒 .
For any particular 𝜋-meson, it is impossible to predict not only the time
of its decay, but also the mode of decay it might “choose”. Instability is
inherent not only in elementary particles, but also in other microparticles.
The phenomenon of radioactivity (spontaneous conversion of isotopes of
one chemical element into isotopes of another, accompanied by emission
of particles) shows that the atomic nuclei can also be unstable. Atoms and
26 bas i c co n c e p ts o f q ua n t u m m e c h a ni cs
Instability determined by the probability laws is, apart from spin, the
second special specific property inherent in microparticles. This may also
be considered as an indication of a certain “internal complexity” in the
microparticles.
𝑛 → 𝑝 + 𝜋 −,
𝑝 → 𝑛 + 𝜋+
microparticles, this idea still holds to a certain extent: the molecules are
made up of atoms, the atoms consist of nuclei and electrons, the nuclei
are made up of protons and neutrons. However, the idea exhausts itself
at this point: for example, “splitting up” of a neutron or a proton does not
reveal the structure of these particles. As regards elementary particles,
when we say that a particle decays into parts, it does not mean that these
particles constitute the given particle. This condition itself might serve as
a definition of an elementary particle.
𝛾 + 𝑝 → 𝑛 + 𝜋 +,
𝛾 + 𝑛 → 𝑝 + 𝜋 −,
𝛾 + 𝑝 → 𝑝 + 𝜋 0,
𝛾 + 𝑛 → 𝑛 + 𝜋 0,
𝛾 + 𝑝 → 𝑝 + 𝜋 + + 𝜋 −,
𝛾 + 𝑛 → 𝑛 + 𝜋 0 + 𝜋 0,
𝛾 +𝑝 →𝑝 +𝑝 +𝑝
It should be mentioned here that in all the above equations, the sum
of the rest masses of the end particles is greater than the rest mass of
the initial ones. In other words, the energy of the colliding particles is
converted into mass (according to the well-known relation 𝐸 = 𝑚𝑐 2 ).
These equations demonstrate, in particular, the fruitlessness of efforts to
break up elementary particles (in this case, nucleons) by “bombarding”
them with other particles (in this case, photons): in fact, it does not lead to
a breaking-up of the particles being bombarded at, but to the creation of
new particles, to some extent at the expense of the energy of the colliding
particles.
conserved.
As a more complicated example, we mention the so-called barionic charge of a particle. It has been
observed that the number of nucleons during an interconversion of particles is conserved. With the
discovery of antinucleons, it was observed that additional nucleons may be created, but they must
he created in pairs with these antinucleons. So a new characteristic of particles, the barionic charge,
was introduced. It is equal to zero for photons, leptons and mesons, +1 for nucleons, and -1 for antin-
ucleons. This permits us to consider the above-mentioned regularity as a law of conservation of the
total barionic charge of the particles. The law was also confirmed by the discoveries that followed: the
hyperons were assigned a barionic charge equal to 1 (as for nucleons) and the antihyperons were given
a barionic charge equal to -1 (as for antinucleons).
While going over from macroparticles to microparticles, one would Universal Dynamic Variables
expect qualitatively different answers to questions like: Which dynamic
variables should be used to describe the state of the object? How should
its motion be depicted? Answers to these questions reveal to a consider-
able extent the specific nature of microparticles.
𝑚 𝑣2
𝐸= + 𝑈 (®
𝑟)
2
𝑝® = 𝑚 𝑣® (1.1)
𝑀® = 𝑚 (®
𝑟 × 𝑣®)
Eliminating the velocity, we get from here the relations connecting en-
ergy, momentum and angular momentum of a classical object:
𝑝2
𝐸= + 𝑈 (®
𝑟) (1.2)
2𝑚
𝑀® = (® ®
𝑟 × 𝑝) (1.3)
𝐸 = ℏ𝜔 (2.3)
frequencies. This had been called the “ultraviolet catastrophe” (see, for Course - Volume 3, volume 3. Mir Publish-
ers, Moscow
example, Volume 3 of Savelyev’s General Physics).4
In 1913, Bohr proposed his theory of the hydrogen atom. This theory The Idea of Quantization and
was evolved as a “confluence” of the planetary atomic model by Ruther- Bohr’s Model of Hydrogen
ford, the Ritz combination principle, and Planck’s ideas of quantization of Atom
energy.
𝐸𝑛 − 𝐸𝑘
𝜔= (2.4)
ℏ
appears in the spectrum. Formula (2.4) expresses the well-known Bohr’s
frequency condition.
(here 𝑚 and 𝑒 are the mass and the charge of an electron, 𝑣𝑛 is the velocity
of the electron in the 𝑛-th orbit). Secondly, Bohr suggested the condition
of quantization of the angular momentum of the electron:
𝑚𝑣𝑛 𝑟𝑛 = 𝑛ℏ (2.5b)
𝑚𝑒 4
𝐸𝑛 = − (2.7)
2ℏ2𝑛 2
The negative sign of the energy means that the electron is in a bound
state (energy of a free electron is taken to be equal to zero).
Substituting the result (2.7) into the frequency relation (2.4), and com-
paring the expression thus obtained with formula (2.2), we may, following
Bohr, find an expression for Rydberg’s constant:
𝑚𝑒 4
𝑅= (2.8)
4𝜋𝑐ℏ3
Bohr’s theory (or the old quantum theory, as it is now called) suffered
from internal contradictions; in order to determine the radius of the orbit,
one had to make use of relations of different kinds - the classical relation
(2.5a), and the quantum relation (2.5b). In spite of this, the theory was
of great significance as a first step towards the creation of a consistent
quantum theory. Moreover, the nature of the spectral terms, and, conse-
quently, the Ritz combination principle, was revealed for the first time
and the calculated value of Rydberg’s constant was in excellent agreement
with its empirical value. The success of the theory proved testimony to
the usefulness of the idea of quantization. Having acquainted himself with
Bohr’s calculations, Sommerfeld wrote Bohr a letter, in which he said:
I thank you very much for sending me your extremely interesting work . . .
The problem of expressing the Rydberg-Ritz constant by Planck’s has been
for some time in my thoughts . . . Although I am for the present still rather
sceptical about atom models in general, nevertheless the calculation of the
constant is indisputably a great achievement.
We must note that in contrast to energy, the angular momentum of a On Quantization of Angular
Momentum
p h y s i c s o f t h e m i c r o pa r t i c l e s 33
𝑀𝑧 = ℏ𝑚 (2.9b)
where 𝑚 = −𝑙, −𝑙 +1, . . . , 𝑙 −1, 𝑙. For a given value of the number 𝑙, the num-
ber 𝑚 can assume 2𝑙 + 1 discrete values. We emphasize here that different
projections of the momentum of a microparticle in a given direction differ
from one another by values which are multiples of Planck’s constant.
In spite of the resounding success of Bohr’s theory, the idea of quan- Anomalies of Quantum Transi-
tization engendered serious doubts in the beginning. It was noticed that tions
the idea was full of internal contradictions. Thus in his letter to Bohr,
Rutherford wrote in 19135 : 5 Bohr, N. (1961). Reminiscences of the
founder of nuclear science and of some
developments based on his work (the
. . .Your ideas as to the mode of origin of the spectrum of hydrogen are very
rutherford memorial lecture, 1958). Proc.
ingenious and seem to work out well; but the mixture of Planck’s ideas Phys. Soc., 78:1083–1115
34 bas i c co n c e p ts o f q ua n t u m m e c h a ni cs
with the old mechanics makes it very difficult to form a physical idea of
what is the basis of it. There appears to me one grave difficulty in your
hypothesis which I have no doubt you fully realise namely, how does an
electron decide what frequency it is going to vibrate at when it passes from
one stationary state to the other? It seems to me that you would have to
assume that the electron knows beforehand where it is going to stop6 . . . 6 The reader should not be confused
by the remarks about the oscillations of
electron: uniform motion in a circle is a
We shall explain the difficulties noticed by Rutherford. Let an electron oc- superposition of two harmonic oscillations
cupy level 𝐸 1 (Figure 2.1). In order to go over to the level 𝐸 2 , the electron in mutually perpendicular directions.
Further contradictions are observed while considering the jump of an electron from one orbit in the
atom to another. Whatever the speed at which the transition of the electron from the orbit of one
radius to that of another takes place, it has to last for some finite period of time (otherwise it would he
a violation of the basic requirements of the theory of relativity). But then it is hard to understand what
the energy of the electron should be during this intermediate period - the electron no longer occupies
the orbit corresponding to energy 𝐸 1 and has not yet arrived at the orbit corresponding to energy 𝐸 2 .
It is thus not surprising that at one time efforts were made to obain
an explanation of experimental results without resorting to the idea of
quantization. In this respect, the famous remarks by Schrödinger about
“these damned quantum jumps”, which, of course, were made in the heat
of the moment, are worth noting.
In this case, there is just one way out: new ideas must be introduced,
which form a non-contradictory picture of the whole including the ideas
of discreteness. The idea of wave-particle duality was just such a new
physical concept.
Classical physics acquaints us with two types of motion: corpuscular Idea of Wave-Particle Duality
and wave motion. The first type is characterized by a localization of the
object in space and the existence of a definite trajectory of its motion. The
second type, on the contrary, is characterized by delocalization in space.
No localized object corresponds to the motion of a wave, it is the motion
p h y s i c s o f t h e m i c r o pa r t i c l e s 35
𝐸 = ℏ𝜔
(2.10)
𝑝=
ℏ𝜔
𝑐
The corpuscular properties of radiation were very clearly demonstrated in the Compton effect (1923).
Suppose a beam of X-rays is scattered by atoms of matter. According to classical concepts, the scat-
tered rays should have the same wavelength as the incident rays. However, experiment shows that
the wavelength of scattered waves was greater than the initial wavelength of the rays. Moreover, the
difference between the wavelengths depends on the angle of scattering. The Compton effect was
explained by assuming that the X-ray beam behaves like a flux of photons which undergo elastic col-
lisions with the electrons of the atoms, in conformity with the laws of conservation of energy and
momentum for colliding particles. This led not only to a qualitative but also to a quantitative agree-
ment with experiment (see Savelyev7 ).
𝐸 = ℏ𝜔
2𝜋ℏ (2.11)
𝑝=
𝜆
The idea of quantization introduces discreteness, and discreteness The Role of Planck’s Constant
requires a unit of measure. Planck’s constant plays the role of such a
measure. It may be said that this constant determines the “boundary”
between microphenomena and macrophenomena. By using Planck’s
constant, as well as mass and charge of an electron, we may form the
following simple composition having dimensions of length:
ℏ2
𝑟1 = = 0.53 × 10−8 cm (2.12)
𝑚𝑒 2
(note that 𝑟 1 is the radius of the first Bohr orbit). According to (2.12), a
p h y s i c s o f t h e m i c r o pa r t i c l e s 37
If the Planck constant ℏ were, say, 100 times larger, then (other con-
ditions being equal) the “limit” of microphenomena would, according to
(2.12), have been of the order of 1 × 10−4 cm. This would mean that the
microphenomena would become much closer to us, to our scale, and the
atoms would have been much bigger. In other words, matter in this case
would have appeared much “coarser”, and classical concepts would have
to be revised on a much larger scale.
Planck’s constant is inseparably linked not only with the idea of quanti-
zation, but also with the idea of duality. From (2.11) it is evident that this
constant plays a fairly important role - it supplies a “link” between the
corpuscular and wave properties of a microparticle. This becomes quite
clear if we rewrite (2.11) in a form permitting us to take account of the
vector nature of momentum:
𝐸 = ℏ𝜔
(2.13)
𝑝® = ℏ𝑘®
Here 𝑘® is the wave vector; its direction coincides with the direction of
propagation of the wave, and its magnitude is expressed through the
wavelength in the following way:
2𝜋
𝑘=
𝜆
The left-hand sides of equations (2.13) describe corpuscular properties of a
microparticle, and the right-hand sides wave properties. We note, by the
way, that the form of relations (2.13) indicates the relativistic invariance of
the idea of duality.
Let us consider an aggregate of a large number of plane waves (the Idea of Duality and Uncer-
nature of waves is not important) propagating, say, along the 𝑥-axis. Let tainty Relations
the frequencies of the waves be “spread” over a certain interval Δ𝜔, and
the values of the wave vector, over an interval Δ𝑘𝑥 . If all these plane
waves are superimposed on one another, we get a wave formation limited
in space called a wave packet (Figure 3.1). The spreading of the wave
packet in space (Δ𝑥) and in time (Δ𝑡) is determined by the relations
Δ𝜔 Δ𝑡 ⩾ 1
(3.1)
Δ𝑘𝑥 Δ𝑥 ⩾ 1
These relations are well known in classical physics. Those acquainted
with radio engineering know that for a more localized signal one must
take more plane waves with different frequencies. In other words, to
reduce Δ𝑥 and Δ𝑡, one must increase Δ𝑘𝑥 and Δ𝜔.
Digressing from the wave packet, we shall formally assume that rela-
tions (3.1) are valid not only for classical waves, but also for wave charac-
teristics of a microparticle. We stress that this assumption by no means
indicates that we shall in fact model a microparticle in the form of a wave
packet. By considering 𝜔 and 𝑘𝑥 in (3.1) as wave characteristics of a mi- Figure 3.1: The wave packet.
Δ𝐸 Δ𝑡 ⩾ ℏ (3.2)
Δ𝑝𝑥 Δ𝑥 ⩾ ℏ (3.3)
These relations were first introduced by Heisenberg in 1927 and are called
uncertainty relations.
Δ𝑝 𝑦 Δ𝑦 ⩾ ℏ, Δ𝑝𝑧 Δ𝑧 ⩾ ℏ, (3.3a)
Δ𝑀𝑦 Δ𝜑 𝑦 ⩾ ℏ, Δ𝑀𝑧 Δ𝜑𝑧 ⩾ ℏ (3.4a)
Let us consider relation (3.3). Here Δ𝑥 is the uncertainty in the 𝑥-coordinate The Meaning of the Uncer-
of the microparticle and Δ𝑝𝑥 the uncertainty in the 𝑥-projection of its mo- tainty Relations
mentum. The smaller Δ𝑥 is, the greater Δ𝑝𝑥 is, and vice versa. If the mi-
croparticle is localized at a certain definite point 𝑥, then the 𝑥-projection
of its momentum must have arbitrarily large uncertainty. If, on the con-
trary, the microparticle is in a state with a definite value of 𝑝𝑥 , then it
cannot be localized exactly on the 𝑥-axis.
(𝐸 2 − 𝐸 1 )𝑇 ⩾ ℏ (3.2a)
Δ𝐸 ⩽ (𝐸 2 − 𝐸 1 ).
𝑇 ⩽ Δ𝑡
Δ𝐸 ≈ 𝐸 2 − 𝐸 1, 𝑇 ≈ Δ𝑡
uncertainty relations.
The method of deriving the uncertainty relations considered in the be- From Diffraction in Microparti-
ginning of this section might appear too formal and unconvincing to some cles to Uncertainty Relations
readers. There are various means of deriving uncertainty relations (see,
p h y s i c s o f t h e m i c r o pa r t i c l e s 41
for example Heisenberg (1930)11 ). One such method [which is specifically 11 Heisenberg, W. (1930). The Physical
Principles of the Quantum Theory. The Uni-
applied to relations (3.3)] is based on a consideration of the phenomena of versity of Chicago Press, Chicago
diffraction of microparticles.
Suppose (Figure 3.2) a screen with a narrow slit is placed in the path
of a strictly parallel beam of certain microparticles with momentum 𝑝.
Let 𝑑 be the width of the slit along the 𝑥-axis (the 𝑥-axis is perpendicular
to the direction of the beam). Diffraction takes place during the passage
of microparticles through the slit. Let 𝜃 be the angle between the initial
direction and the direction of the first (principal) diffraction peak. The
classical wave theory gives the following well-known relation for this
angle:
𝜆
sin 𝜃 =
𝑑
Assuming angle 𝜃 to be sufficiently small, we can rewrite this relation in
the following form:
𝜆
𝜃≈ (3.5)
𝑑
If by 𝜆 we now mean not the classical wavelength, but the length of the
de Broglie wave (i.e. the wave characteristic of the microparticle, we may
rewrite relation (3.5) in “corpuscular language” by using the expression
(2.11):
ℏ Figure 3.2: Diffraction of microparticles
𝜃≈ (3.5a) from a slit.
𝑝𝑑
But how we are to understand the existence of the angle 𝜃 in “corpuscular
language”? Obviously, it means that while passing through the slit, the
microparticle acquires a certain momentum Δ𝑝𝑥 in the direction of the
𝑥-axis. It is easy to see that Δ𝑝𝑥 ≈ 𝑝𝜃 . Substituting (3.5a) into this, we get
ℏ
Δ𝑝𝑥 ≈
𝑑
By considering the quantity 𝑑 as the uncertainty Δ𝑥 in the 𝑥-coordinate of
the microparticle passing through the slit, we get
Δ𝑝𝑥 Δ𝑥 ≈ ℏ,
i.e. we arrive at the uncertainty relation (3.3). Thus the attempt to de-
termine in some way the coordinate of a microparticle in a direction
perpendicular to the direction of its motion leads to an uncertainty in the
momentum of the microparticle in that direction, which also explains the
phenomenon of diffraction observed in the experiment.
In order to describe the state of a classical object it is necessary to Uncertainty Relations and the
give a definite set of numbers - the coordinates and the velocity compo- State of Microparticles. The
nents. In doing this other quantities, in particular, energy, momentum Concept of a Complete Set of
and angular momentum of the object will also be determined [see (1.1)] Physical Quantities
The uncertainty relations show that this method of defining a state is
42 b a sic co n c e p t s o f q uan t u m m e c h a n ics
Each group contains the states of the microparticle in which the values of
the corresponding complete sets are known (it is customary to say that
every complete set has its own method of defining its states).
𝑥, 𝑦, 𝑧, 𝜎 (3.6a)
𝑝𝑥 , 𝑝 𝑦 , 𝑝𝑧 , 𝜎 (3.6b)
𝐸, 𝑙, 𝑚, 𝜎 (3.6c)
(remember that 𝑙, 𝑚, and 𝜎 are orbital, magnetic and spin quantum num-
bers, respectively). We emphasize that the coordinates and the momen-
tum components of a microparticle (in this case an electron) fall in differ-
ent complete sets of quantities; these two physical quantities cannot be
measured simultaneously. Hence the classical relations (1.2) and (1.3) are
not valid when going over to microparticles, since each of these relations
contains the coordinates as well as the momentum.
(𝑝𝑥2 + 𝑝 𝑦2 + 𝑝𝑧2 )
𝐸=
2𝑚
The set (3.6c) is usually employed for describing the states of an electron
in the atom. To describe the states of a photon, the following sets are most
commonly employed:
𝑘 𝑥 , 𝑘 𝑦 , 𝑘𝑧 , 𝛼 (3.7a)
𝐸, 𝑀 , 𝑀𝑧 , 𝑃
2
(3.7b)
The set (3.7a) is used for describing the states of photons correspond-
ing to plane classical waves, in this case the energy of the photon is also
defined (recall that 𝐸 = ℏ𝜔 = ℏ𝑐𝑘). The states described by the set (3.7a)
are called 𝑘𝛼-states. The set (3.7b) is employed for describing the states
of photons belonging to spherical classical waves. We note that just as
a spherical wave may be represented as a superposition of plane waves,
the states of a photon described by the set (3.7b) may be represented as
a “superposition” of states described by the set (3.7a). The converse state-
ment regarding the representation of plane waves as a superposition of
spherical waves is also true. Here we have touched upon (for the present
just touched upon) one of the most important and delicate aspects of the
quantum-mechanical description of matter - the specific character of the
“interrelations” between states of a microparticle described by different
complete sets. This specific character is reflected in the most constructive
principle of quantum mechanics - the principle of superposition of states.
The superposition of states will be considered in detail in the second chap-
ter; here we shall just restrict ourselves to the above-mentioned remarks.
The main contradiction regarding quantum transitions indicated in Sec- The Uncertainty Relations and
tion 2 is essentially overcome by making use of the idea of duality or, Quantum Transitions
more precisely, the uncertainty relation (3.2). Let us consider transition of
an electron in an atom from level 𝐸 1 to level 𝐸 2 by absorbing a photon of
44 b a sic co n c e p t s o f q uan t u m m e c h a n ics
The uncertainty relation (3.2) allows us to introduce and employ a very important concept in quan-
tum theory, the so-called virtual transitions, for explaining quantum transitions. We shall give here a
simplified treatment of virtual transitions, but we shall give a detailed explanation later in Section 6.
According to relation (3.2), an electron may go over from level 𝐸 1 to 𝐸 2 without getting any energy
from outside; what is important is that it should quickly return to its initial level 𝐸 1 . Such a “journey”
(𝐸 1 → 𝐸 2 → 𝐸 1 ) is possible if its duration Δ𝑡 is such that the inequality ℏ/Δ𝑡 (𝐸 2 − 𝐸 1 ) is satisfied,
because in this case the uncertainty in the energy of the electron is greater than the difference in the
energies of the levels under consideration. Hence it is clear that the statement “the electron occupies
level 𝐸 1 ” may be understood quite specifically - as incessant “transition” of the electron from the given
state to others with an inevitable return every time to the starting level 𝐸 1 . Such transitions cannot he
observed experimentally, and are called virtual transitions in contrast to the normal (real) transitions.
During interaction of an electron undergoing virtual transitions with radiation, the electron is liable
to change its “residence”. For example, it might now occupy level 𝐸 2 and will in future perform virtual
transitions not from level 𝐸 1 , but from level 𝐸 2 . If such a thing happens, the electron is said to have
absorbed a photon of energy ℏ𝜔 = 𝐸 2 − 𝐸 1 , and undergone a transition from level 𝐸 1 to 𝐸 2 . Virtual
transitions don’t require any expenditure of energy from outside while a real transition cannot occur
without expenditure of energy - the energy of the photons absorbed (or emitted) by electrons during
interaction with radiation.
To explain the difference between real and virtual transitions, we note that a real transition from a
level 𝐸 1 to another level 𝐸 2 and back may be broken up into two successive events in time (in between
the electron may be experimentally registered in the intermediate state 𝐸 2 ). However, the virtual
transition from level 𝐸 1 to 𝐸 2 and back cannot be broken up into two events in time - both parts of the
transition must be considered as a single, indivisible process in time.
The uncertainty relations used in quantum theory are by no means ex- Uncertainty Relation “Number
hausted by relations (3.2)-(3.4). As an example of one more such relation of Photons-Phase”
we consider the uncertainty relations for the number of photons and the
p h y s i c s o f t h e m i c r o pa r t i c l e s 45
Δ𝐸 = ℏ𝜔Δ𝑁
𝑝2 𝑒2
𝐸= − (4.1)
2𝑚 𝑟
where 𝑚 and 𝑒 are the mass and charge of the electron, respectively. In
46 b a sic co n c e p t s o f q uan t u m m e c h a n ics
order to use the classical expression (4.1) in the quantum theory, we con-
sider the quantities 𝑝 and 𝑟 occurring in it as uncertainties in momentum
and coordinates of the electron, respectively. According to relation (3.3),
these quantities are connected with each other. We assume 𝑝𝑟 ≈ ℏ, or,
simply,
𝑝𝑟 = ℏ (4.2)
𝑝 2 𝑒 2𝑝
𝐸 (𝑝) = − (4.3)
2𝑚 ℏ
It is easy to see that the function 𝐸 (𝑝) has a minimum for a certain value
𝑝 = 𝑝 1 . We denote it as 𝐸 1 . The quantity 𝐸 1 may be considered as the
energy of the ground state of the hydrogen atom while the quantity 𝑟 1 =
ℏ
is the estimate of the linear dimensions of the atom (in Bohr’s theory
𝑝1
𝑑
this is the radius of the first orbit). By equating the derivative 𝐸 (𝑝) to
𝑑𝑝
𝑚𝑒 2
zero, we find 𝑝 1 = . This at once gives the required evaluations (cf.
ℏ
(2.6) and (2.7)]:
ℏ2
𝑟1 =
𝑚𝑒 2
(4.4)
𝑚𝑒
4
𝐸1 = − 2
2ℏ
The values given by (4.4) fully coincide with the results of the rigorous
theory.14 Of course such a complete coincidence must be considered to 14 In the rigorous theory the quantity
𝑟 1 is a characteristic for the ground state
some extent as an accidental success. Only the order of the quantities of the hydrogen atom and denotes the
should be taken seriously here. We emphasize that this order can be distance from the nucleus at which an
electron is most likely to be observed (see
evaluated quite simply as follows: it is sufficient first to simply replace the expression (5.4)).
precise values of the dynamic variables in expression (4.1) by quantities
which characterize the degree of “blurring” of these variables, i.e. by their
uncertainties, and then use the quantum-mechanical relations connecting
the said uncertainties.
We shall proceed exactly in the same way as in the preceding example. Estimate of the Energy of
The energy of a classical one-dimensional harmonic oscillator is given by Zero-point Oscillations of an
the expression Oscillator
𝑝 2 𝑚𝜔 2𝑥 2
𝐸= 𝑥 + (4.5)
2𝑚 2
Treating 𝑝𝑥 and 𝑥 as uncertainties in the momentum and coordinate of
the oscillating microparticle and using the equality 𝑝𝑥 · 𝑥 = ℏ as the
uncertainty relation, we get from (4.5)
𝑝𝑥2 𝑚𝜔 2 ℏ2
𝐸 (𝑝𝑥 ) = + (4.6)
2𝑚 2𝑝𝑥2
𝑑
By equating the derivative 𝐸 (𝑝𝑥 ) to zero we find the value of 𝑝 0 =
√ 𝑑𝑝
± 𝑚ℏ𝜔 for which the function 𝐸 (𝑝𝑥 ) assumes its minimum value. It is
p h y s i c s o f t h e m i c r o pa r t i c l e s 47
𝐸 0 = 𝐸 (𝑝 0 ) = ℏ𝜔 (4.7)
The essence of the effect, investigated in 1958 by Keldysh, and inde- Evaluation of the “Blurring” of
pendently by Franz, lies in the following: in a uniform external electric the Optical Absorption Band
field, the minimum of the electron energy in the conduction band of semi- Edge in the Franz-Keldysh
conductors shifts downwards on the energy scale, leading to a “blurring” Effect
of the edge of the fundamental optical absorption band (as a result the
absorption of the photons with energies lower than the forbidden band
width becomes possible).15 The value of the shift of electronic states char- 15 Vavilov, V. S. (1963). The Effect of Ra-
diation on Semiconductors (in Russian). Fiz-
acterizing the indicated “blurring” may be obtained in the same way as matgiz, Moscow
the preceding evaluations were made. We make use of the classical expres-
sion for the energy of a charged particle in the electric field of intensity E :
𝑝𝑥2
𝐸= − E 𝑒𝑥 (4.8)
2𝑚
Here 𝑚 is the effective mass of the electron in the conduction band. Treat-
ing 𝑝𝑥 and 𝑥 as uncertainties in the momentum and coordinate of the
electron and using the equality 𝑝𝑥 · 𝑥 = ℏ. as the uncertainty relation, we
get from (4.8)
𝑝𝑥2 E 𝑒ℏ
𝐸 (𝑝𝑥 ) = − (4.9)
2𝑚 𝑝𝑥
𝑑
Next, as usual we equate the derivative 𝐸 (𝑝𝑥 ) to zero and obtain the
𝑑𝑝𝑥
48 b a sic co n c e p t s o f q uan t u m m e c h a n ics
√3
value 𝑝 0 = − E 𝑒ℏ𝑚 for which the function 𝐸 (𝑝𝑥 ) assumes its minimum
value:
r
3 3 (E 𝑒ℏ) 2
𝐸0 =
2 𝑚
r (4.10)
3 (E 𝑒ℏ)
2
≈
𝑚
Expression (4.10) gives an estimate of the extent of the “blurring” of the
edge of the fundamental optical absorption band in the Franz-Keldysh
effect.
While postulating the stationary states, Bohr’s theory did not explain Why does not the Electron Fall
why, after all, the electron, moving under acceleration, does not radiate into the Nucleus?
and fall into the nucleus as a result of this. Relation (3.3) explains this
fact. The falling of an electron into a nucleus would obviously mean a
considerable reduction in the uncertainty of its coordinate. Before the
hypothetical falling into the nucleus, the electron is localized within
the limits of the atom, i.e. in a region of space with linear dimensions
ℏ2 /𝑚𝑒 2 ≈ 1 × 10−8 cm [see (4.4)], whereas after falling into the nucleus
it would be localized in a region with linear dimensions less than 1 ×
10−12 cm. According to (3.3) a stronger localization of a microparticle in
space is linked with a “blurring” of its momentum. Hence upon falling
into the nucleus, the mean value of the momentum of the electron must
increase, which requires an expenditure of energy. Thus it turns out
that effort has to be made not to “hold” the electron from falling into the
nucleus, but on the contrary to “force” the electron to be localized within
the nucleus.
In the example of the zero-point oscillations it was pointed out that the
microparticle in a potential well always possesses a non-zero minimum
energy 𝐸 0 . The magnitude of 𝐸 0 depends, in particular, on the spatial
dimensions of the well (or on its width 𝑎, which determines the extent
of localization of the microparticle in space). By taking into account the
uncertainty relation, it is easy to see that
ℏ2
𝐸0 ≈ (4.11)
𝑚𝑎 2
If 𝑎 decreases, 𝐸 0 increases. For sufficiently small 𝑎, the energy 𝐸 0 may
become greater than the depth of the potential well. It is obvious that
such a well will not hold the microparticle at all.
energy of the electron turns out to be a few orders higher than the bind-
ing energy of a nucleon in the atomic nucleus (the latter being not greater
than 1 × 107 eV). This means that the electron cannot ever be present in
the nuclear potential well and hence it can by any means be compelled
to be localized within the nucleus, not even by force. This not only elimi-
nates the problem of “an electron falling into the nucleus” but also solves
another fundamental question: the electron is not one of the constituents of
the atomic nucleus.
In order to draw the trajectory of a particle, it is necessary, strictly On the “Trajectory” of Mi-
speaking, to know the coordinate and momentum of the particle at every croparticles
moment of time (in fact, in order to depict the dependence 𝑥 (𝑡), it is
necessary to know, for every 𝑡, the values 𝑥 and 𝑑𝑥/𝑑𝑡). Since, according
to the uncertainty relation (3.3), a microparticle cannot simultaneously
possess a defInite coordinate and a definite projection of the momentum,
one can draw the conclusion that the concept of trajectory in case of
microparticle, strictly speaking, is not applicable.
We note that with the rejection of the idea of orbits of the electron in an atom the contradiction re-
garding the problems of the instantaneous jump of the electron from one orbit into another, discussed
in Section 2, is automatically eliminated.
There are situations, however, in which one can make use of the idea
of “the trajectory of a microparticle”. As an example we consider the
motion of electrons in the kinescope of a television set. The momentum
√
of the electron along the axis of the tube is 𝑝 = 2𝑚𝑒𝑈 , where 𝑈 is the
50 bas i c co n c e p ts o f q ua n t u m m e c h a ni cs
Suppose we have a potential barrier whose height 𝑈 is greater than The Possibility of a Microparti-
the energy of the particle (Figure 4.1). We ask the question: can a particle, cie Sub-Barrier Passage (Tun-
situated somewhere to the left of the barrier, appear after some time to the neling Effect)
right of it, without getting any energy from outside? Classical mechanics
gives a negative answer – a classical corpuscle cannot pass through the
barrier. If this were to happen, then the total energy of the particle, say, at
the point 𝐴 in Figure 4.1 would be less than its potential energy, which is
physically absurd.
It has been noted above that the energy of a freely moving microparticle is not quantized. This may be
easily shown by making use of the tunneling effect. Suppose a microparticle is located in a potential
well shown in Figure 4.2. On account of the tunneling effect the microparticle may of its own accord
leave the potential well. Consequently, the time for which it stays in the well is not infinite. If we de-
note this time as Δ𝑡, it follows from (3.2) that the energy of the microparticle must have an uncertainty
of the order ℏ/Δ𝑡. We reduce the width 𝑏 of the potential barrier (dotted line in Figure 4.2). It is clear
that as a result the magnitude of Δ𝑡 will decrease, since the probability of the rnicroparticle leaving
the well will increase. With a decrease in Δ𝑡, the uncertainty in the energy of the microparticle, ℏ/Δ𝑡,
will increase. This may be considered as a larger blurring (further broadening) of the energy levels of
the rnicroparticle in the well. In the limiting case of zero thickness of the barrier, the value of Δ𝑡 van-
ishes, the microparticle becomes a freely moving particle, and the energy levels broaden up indefinitely,
actually transforming into a continuous energy spectrum.
The process of “breaking up” of objects surrounding us into smaller A Microparticle Is not a Classi-
and smaller “fractions” leads to microparticles. Therefore it is but natural cal Corpuscle
to associate microparticles first of all with corpuscles. This is also sup-
ported by the fact that a microparticle is characterized by a definite rest
mass and a definite charge. For instance, it is meaningless to speak of a
half-electron having half the mass and half the electric charge of a whole
electron. The very terms “microparticle” and “elementary particle” reflect
the notion of the microparticle as being some particle (corpuscle).
The analysis of one mistake which is committed quite often even Microparticle Is Not a Classi-
these days when considering a simplified account of quantum mechan- cal Wave
ics is quite instructive. We shall demonstrate this mistake through two
examples.
𝑝𝑛 𝑟𝑛 = 𝑛ℏ (5.1)
It is stated that the wave properties of an electron permit a very simple Example 2
derivation of the formula for the energy levels in a potential well, if we
assume that a definite number of de Broglie half-waves are confined in the
potential well (in analogy with the number of half-waves contained in the
length of a string fixed at both ends) corresponding to different stationary
states. Designating the width of the one-dimensional potential well by 𝑎,
𝑛𝜆𝑛
we write 𝑎 = , from which we get the desired result:
2
𝑛 2 𝜋 2 ℏ2
𝐸𝑛 = (5.2)
2𝑚𝑎 2
Both these final results [(5.1) as well as (5.2)] are correct; they are the
same as the result deduced from strict theory. However, the “derivation”
of these formulas must be considered to be unsound. In both cases in fact
the same fundamental mistake has been committed: they are based on the
wrong assumption that the electron in a potential well has a definite de
Broglie wavelength, or, in other words, a definite momentum. However,
according to (3.3), the momentum of a microparticle in a bound state is
characterized by the uncertainty Δ𝑝 ⩾ ℏ/𝑎. Since in the above examples
𝑝 ≈ ℏ/𝜆 ≈ ℏ/𝑎, it follows that the momentum is of the same order of
magnitude as the uncertainty in momentum given by relation (3.3). It is
clear that in such cases one cannot speak about any value of the electron
momentum (and correspondingly of its de Broglie wavelength) even
approximately.17 These examples demonstrate an obvious exaggeration 17 This question is considered in greater
detail in Section 23 of this book. See also
work by de Broglie.
de Broglie, L. (1930). Introduction a l’
Etude de la Méchanique Ondulatoire (in
French). Hermann, Paris
p h y s i c s o f t h e m i c r o pa r t i c l e s 53
If a microparticle is neither a corpuscle nor a wave, then may he it Attempts to Represent a Mi-
is some kind of a symbiosis of a corpuscle and a wave? Several attempts croparticle as a Symbiosis of a
were made to model such a symbiosis and thus also to visually demon- Corpuscle and a Wave
strate the wave-particle duality. One such attempt represents a micropar-
ticle as a formation, limited in space and in time. This may be the wave
packet mentioned in Section 3. This may also be just a “scarp” of a wave,
often called a wave train. Another attempt uses a model of a pilot-wave,
according to which a microparticle is some sort of a “compound” of a
18 Bohm, D. (1957).
corpuscular “core” with a certain wave which controls the motion of the Causality and
Chance in Modern Physics. Van Nostrand,
core. N.Y
One of the versions of the pilot wave model is considered by D. Bohm in his book 18 : We first postu-
late that connected with each of the “fundamental” particles of physics (e.g. an electron) is a body existing
in a small region of space . . . in most applications at the atomic level the body can be approximated as
a mathematical point . . . The next step is to assume that associated with this body there is a wave with-
out which the body is never found. This wave will be assumed to be an oscillation in a new kind of field,
which is represented mathematically by the 𝜓 -field of Schrödinger somewhat like the gravitational and the
electromagnetic, but having some new characteristics of its own . . .
We now assume that the 𝜓 -field and the “body” are interconnected in the sense that the 𝜓 -field exerts a
new kind of “quantum-mechanical” force on the body, the force is such as to produce a tendency to pull the
body into regions where |𝜓 | is largest.
If the above tendency were all that were present, the body would eventually find itself at the place where
the 𝜓 -field had the highest intensity. We now further assume that this tendency is resisted by random
motions undergone by the body, motions which are analogous to the Brownian movement. They could, for
example, come from random fluctuations in the 𝜓 -field itself.
Once admitting the existence of these fluctuations, we then see that they will produce a tendency for
the body to wander in a more or less random way over the whole space accessible to it. But this tendency
is opposed by the “quantum force” which pulls the body into the places where the 𝜓 -field is most intense.
The net result will be to produce a mean distribution in a statistical ensemble of bodies, which favours the
regions, where the 𝜓 -field is most intense . . .
Figure 5.1 illustrates the given model as applied to the passage of a microparticle through the
screen with slits: the 𝜓 -wave is diffracted on both the slits, while the “body” passes through one slit
and is registered on the screen in accordance with the result of the interference of 𝜓 -waves.
54 bas i c co n c e p ts o f q ua n t u m m e c h a ni cs
It is not denied that such models could appear attractive at first glance
if only because of their intuitive appeal. It must be emphasized at once,
however, that all these models are baseless. We shall not explain at this
stage the reasons for worthlessness of the pilot-wave model considered
above, but shall just mention that it is cumbersome since it uses artificial
notions such as the 𝜓 -field which is “to some extent similar to gravita-
tional end electromagnetic fields”, or the “quantum force” which reflects
the interaction of certain “object” with the 𝜓 -field. The reader will later re-
alize that the worthlessness of such models is not because of some specific
feature, but because of deep fundamental reasons. He will understand that
any attempt at a literal interpretation of the wave-particle duality, any
attempt to model a symbiosis of corpuscle and a wave, should be consid-
ered fruitless from the very start. A microparticle is not a symbiosis of a
corpuscle and a wave.
Figure 5.1: Bohm’s pilot wave model.
At present the wave-particle duality is considered as the potential How to Understand the Wave-
ability of a microparticle to exhibit its different properties depending on Particle Duality
external conditions, in particular, on the conditions of observation. As
Fock wrote19 : 19 Fock, V. A. (1957). On the interpreta-
tion of quantum mechanics. Czechoslovak
Journal of Physics, 7:643–656
Thus under certain conditions an atomic object may exhibit wave prop-
erties and under other conditions corpuscular properties; conditions are
also possible when both kinds of properties appear simultaneously but not
sharply. We can state that it is potentially possible for an atomic object to
manifest itself either as a wave or as a particle or in an intermediate fash-
ion, according to the external condition prevailing. It is just this potential
possibility of exhibiting various properties inherent in an atomic object that
constitutes the wave-corpuscular duality. Any other, more literal mean-
ing attached to this duality, such as a wave-particle model of any kind, is
incorrect.
The absence of a visual model of a microparticle does not in any way Electron in an Atom
prevent us from using tentative models quite suitable for representing a
microparticle under different conditions. As an example, let us consider
an electron in an atom.
In order to describe the dimensions and the form of the electron cloud,
we introduce a certain function
In Figure 5.2 (a) are shown forms of functions 𝑤𝑛𝑙 (𝑟 ) for different states
of the electron in a hydrogen atom. Notice that the functions 𝑤 10, 𝑤 21, 𝑤 32
have maxima corresponding to the radii of first, second and third orbits in
Bohr’s theory. Figure 5.2 (b) shows forms of function 𝑍𝑙𝑚 for some states
of the electron. For 𝑙 = 0 (for the so-called 𝑠-electron) we have a spherical
56 bas i c co n c e p ts o f q ua n t u m m e c h a ni cs
The ground state of the hydrogen atom is characterized by a spherical Quantum Mechanics. D. Reidel Publishing
Co
electron cloud. Theory shows (see, for example, Blokhintsev 20 ) that in
this case
𝑟2 2𝑟
𝑤𝑛𝑙 (𝑟 ) = 4 exp − (5.4)
𝑟 13 𝑟1
The parameter 𝑟 1 characterizing the effective radius of the cloud is deter-
mined by relation (4.4); in Bohr’s theory it occurs as the radius of the first
orbit.
The rejection of the classical individualization of an object is quite fun- Identity of Microparticles
damental. In classical mechanics objects are known to have individuality
since it is always possible in principle to enumerate them and observe
the behaviour of anyone of them. In this case, however alike two classical
objects may be, they are never identical and can always be distinguished.
But in quantum mechanics two microparticles of the same type should be
treated as absolutely identical. Thus, all electrons are identical and so are
all unexcited hydrogen atoms, helium nuclei, etc.
Suppose we have several electrons, one of which is “assigned” the number 1 at the moment of time
𝑡 = 0. Can this electron be identified after a certain time 𝑡? Such an identification could have been
easily done if we could put some “label” on this isolated object. We could get by without “labelling”
this electron if we could simply keep a watch over the isolated object, i.e. if we could “mentally” follow
it (in our imagination) along its trajectory. This is precisely what we would have done in the case
of isolated classical objects. However, none of this holds in the case of an electron; it is in principle
impossible to “label” it. Strictly speaking, it has no trajectory. The electron “isolated” by us at the
instant 𝑡 = 0, cannot be isolated in actual practice: it does not have the individuality which would
allow it to be identified in the assembly of electrons after a certain time 𝑡: Two electrons are much
more “like each other” than the proverbial “two peas in a poll”; since the latter are classical objects,
they could differ in size or in chemical composition in some way.
Laplacian determinism excludes the element of chance from the be- Chance and Necessity in the
haviour of an isolated object; in classical mechanics, necessity completely Behaviour of a Microparticle
dominates. Because of this, the laws of classical mechanics are dynamic
and not statistical. The element of chance (and, consequently, statistical
laws also) appear in classical physics only when considering aggregates of
objects or assemblies of particles.
with probability as one of its basic attributes. As Fock 21 has remarked, 21 Fock, V. A. (1957). On the interpreta-
tion of quantum mechanics. Czechoslovak
“in quantum physics the concept of probability is a primary concept and Journal of Physics, 7:643–656
plays a fundamental role.” It could be said that the behaviour of an in-
dividual microparticle is random, but the probability of this behaviour
is necessary.22 The electronic cloud considered in Section 5 may serve 22 Here, it is quite appropriate to recall
the words of F. Engels: Necessity emerges
as a good example of this. The occurrence of an electron at some point from within the framework of randomness.
near the nucleus is a random event, but the probability of its being found
at a given point (𝑟, 𝜃, 𝜑) is definite - it is described by a function of the
type (5.3) or, in other words, is determined by the shape and size of the
corresponding electron cloud.
Perhaps there is nothing more alien to classical physics than the idea Virtual Transitions and Virtual
of virtual transitions and virtual microparticles. The virtual transition Microparticles
of an electron from level 𝐸 2 to level 𝐸 1 and back (the transition 𝐸 2 →
𝐸 1 → 𝐸 2 ) may be considered as a process in which the electron emits and
absorbs a photon of energy (𝐸 2 − 𝐸 1 ). Such a photon is called virtual. In
contrast to the photons participating in real transitions, virtual photons
cannot be observed experimentally. The creation of a virtual photon is not
connected with an absorption of energy from outside, and its annihilation
is not connected with a release of energy. The law of conservation of
energy is not violated since a virtual photon exists for a very short time
Δ𝑡 and, according to uncertainty relation (3.2), the energy of an electron
ℏ
emitting the virtual photon is characterized by the uncertainty Δ𝐸 ≥ ,
Δ𝑡
which may be of the order of, or greater than, the energy of the photon
(𝐸 2 − 𝐸 1 ). The emission or absorption by an electron of virtual photons
corresponds, from a physical point of view, to the process in which an
electron undergoes virtual transitions.
Let us consider the diagram (a). Here 1 and 2 are the electrons before
interaction with each other (before scattering), 𝐴𝐵 is a virtual photon
which is exchanged by the electrons during the process of interaction
(note that all the particles indicated in the diagram by lines connecting
two vertices are virtual); 3 and 4 are the electrons after scattering. Let us
turn to the diagram (𝑏). Here 1 and 2 are electrons before scattering, 𝐴𝐵
and 𝐶𝐷 are virtual photons exchanged by the electrons, 3 and 4 are virtual
electrons, 5 and 6 are electrons after scattering. The diagram (c) is of the
same type as diagram (b); here the electrons exchange two photons. The
diagram (d) shows one of the processes in which the electrons exchange
three photons. It is obvious that there is an infinite number of such dia-
grams which become more and more complicated (with the participation Figure 6.1: Scattering of electrons as
of more and more photons). shown in Feynman diagrams.
lated not to quantum mechanics, but to quantum field theory (quantum Feynman’s diagrams is given, for example,
in (Cooper, 1968).
23
electrodynamics) . However, a general introduction to the ideas form- Cooper, L. N. (1968). An Introduction
to the Meaning and Structure of Physics.
Harper and Row, N.Y
p h y s i c s o f t h e m i c r o pa r t i c l e s 61
Before ending this discussion on Feynman’s diagrams, we consider the Figure 6.2: Virtual creation of electron-
so-called effect of polarization of a vacuum. Figure 6.2 shows a diagram positron from a photon.
describing one of the processes responsible for this effect. A photon is
transformed into a virtual electron-positron pair, which is annihilated
and transformed again into a photon (one of the solid lines between
the vertices of the diagram “shows” a virtual electron, and the other, a
virtual positron). The members of this pair during their lifetime may ob-
viously generate virtual photons and, consequently, new virtual electron-
positron pairs, and so on. As a result of this, the vacuum turns out to be
not “empty” but “filled” with virtual electric charges which must exercise
a screening effect on external (real) charges. Experimental confirmation of
this effect is the best evidence of the usefulness of our concept of virtual
particles.
As we have already mentioned, the existence of the element of chance The Microparticle and Its
in the behaviour of a microparticle is one of its most specific properties. Surroundings
As a result of this, quantum mechanics turns out in principle to be a
statistical theory operating with probabilities. But what is the reason for
the existence of an element of chance in the behaviour of a microparticle?
During the interaction of an electron with photons there is, strictly speak-
ing, no electron and no photon but a single entity which must be consid-
ered as such without going into details.
It seems necessary to give up the idea that the world can correctly be anal-
ysed into distinct parts, and to replace it with the assumption that the
entire universe is basically a single, indivisible unit. Only in the classical
limit can the description in terms of component parts be correctly applied
without reservations. Wherever quantum phenomena play a significant
role, we shall find that the apparent parts can change in a fundamental
way with the passage of time, because of the underlying indivisible connec-
tions between them. Thus, we are led to picture the world as an indivisible,
but flexible and everchanging, unit.25 25 Bohm, D. (1951). Quantum Theory.
Prentice-Hall, N.Y
p h y s i c s o f t h e m i c r o pa r t i c l e s 63
aut hor: The impossibility of the classical interpretation of a mi- Participants: (same as in Pre-
croparticle predetermines a negative answer to the question “Is it lude).
possible to have a “physically intuitive” model for a microparticle?”
aut hor: There are very sound reasons for this. I shall indicate just
two of them. Firstly, any modelling envisages in the long run a
detailization irrespective of whether it is a model of an object or a
process. However, the impossibility of an unlimited detailization
is characteristic of microparticles and microphenomena, as we
have already mentioned. This important situation was persistently
stressed by Bohr. He wrote, in particular (see his article Quantum
Physics and Philosophy 26 ): 26 Bohr, N. (1963). Essays 1958-1962 on
atomic physics and human knowledge. In-
terscience, N.Y., Interscience, NY
A new epoch in physical science was inaugurated, however, by
Planck’s discovery of the elementary quantum of action, which
revealed a feature of wholeness inherent in atomic divisibility of
matter. Indeed, it became clear that the pictorial description of
classical physical theories represents an idealization valid only
for phenomena in the analysis of which all actions involved are
sufficiently large to permit the neglect of the quantum . . .
aut hor: Of course, this is not true. First of all, you overlook the
fact that the electron has quite definite characteristics like rest mass,
electric charge, spin, etc. It is stable and is a fermion. As regards
a “physically intuitive” model of an electron, well, it simply does
not exist. However, in rejecting a “physically intuitive” model of a
microparticle, quantum mechanics in no way sacrifices objectivity
in favour of subjectivity. It is just that the electron is a very compli-
cated physical object, and depending on the external circumstances,
including circumstances of observation, it exhibits its different as-
pects, which objectively existed in potential form (I stress this) even
before the observer was born. A sober assessment of this complex
situation is that a “physically intuitive” model of the electron is not
possible.
aut hor: I don’t agree that we don’t even know what an electron
is. I have just indicated a number of precisely determined charac-
teristics and properties of an electron. More detailed properties of
microparticles in general and electrons in particular were consid-
ered in the preceding sections of the book (and will be considered in
the following sections). In fact, we know quite a lot about the elec-
tron and know, in particular, about its behaviour in a crystal. This is
evidenced by the large number of semiconducting devices fabricated
and used by us in practice. As you will see, the absence of a “phys-
p h y s i c s o f t h e m i c r o pa r t i c l e s 65
aut hor: I can understand your doubts. For you, apparently, only
the extremes matter: either graphic models, or mathematical ab-
straction. To you, either a model should reflect everything or almost
everything, otherwise it is quite useless. The doubts arising in your
mind are a consequence of precisely this type of viewpoint. How-
ever, the quantum-mechanical approach to such questions is more
flexible, or rather, dialectical.
is used quite extensively and flexibly. Moreover, all models are not
interpreted literally but tentatively.
Quantum Mechanics
ph y s i c al f o u n dat i o n s o f q ua n t u m m e c h a n i c s 69
The concepts of quantum mechanics are based on a vast collection of Actual Experiments and the
experimental data gathered over a period of more than fifty years at the System of Basic Experiments
th th
end of the 19 century and in the first half of the 20 century. Among
the large number of experiments, a few stand out as being definite “mile-
stones” and can hence be called decisive. They include the experiments of
Lummer and Pringsheim on black body radiation coupled with Planck’s
theoretical works (1900), the experiments of Frank and Hertz (1914) on in-
elastic collision of electrons with atoms, Millikan’s experiments (1914) on
the photoelectric effect, confirming the laws predicted earlier by Einstein,
the experiments conducted by Stern and Gerlach (1921) on the splitting
of atomic beams in non-uniform magnetic fields, the measurements of
wavelengths of X-rays scattered by matter carried out by Compton (1923),
and the experiments of Davisson and Germer, and Tartakovsky (1927)
on electron diffraction.1 These experiments (and many others which did 1 A description of these experiments
may be found, for example in Trigg.
not become so famous) constitute the foundation on which, over a num-
Trigg, G. L. (1971). Crucial Experiments
ber of decades, quantum theory was built, perfected, freed from various in Modern Physics. Van Nostrand, N.Y
paradoxes, and finally brought to its present harmonious structure.
Looking now from the position of the existing quantum theory at the
experimental quest which led to it, it is worth generalizing the actual
experimental picture by omitting the details that do not play a signifi-
cant role and trying to conceive the simplest system of basic experiments
which describe the fundamental aspects of the quantum-mechanical view-
point. In this section an attempt has been made to consider such a system
of experiments. This system is built on the basis of actual experiments but
one should not look for a one-to-one correspondence between the basic
experiments and actual experiments conducted at a certain time in a cer-
tain laboratory. Basic experiments must be seen as a sort of generalization
of several actual experiments. Hence, the experimental details concerning
a certain apparatus or various details of a historical nature do not play a
significant role here.
Let us begin by considering the well-known experiment on the in- Experiment 1 (Microparticles
terference of light waves. Figure 7.1 schematically shows the simplest in an Interferometer)
interferometer. Here, 1 is a point source of monochromatic light, 2 is a
screen with two small slits 𝐴 and 𝐵, and 3 is a detector screen registering
the intensity of light impinging it. This intensity indicated on the diagram
by the curve 𝐼 (𝑥). The interference character of the curve 𝐼 (𝑥) is fairly
simply explained within the framework of classical wave theory of light:
the light wave from source 1 upon reaching is the screen 2 converts the
slits 𝐴 and 𝐵 into sources of new light waves, which add up to give on
screen 3 the characteristic interference pattern of intensity distribution.
One might ask what relation can the phenomenon of the interference
of light, discovered and explained long ago, have with quantum mechan-
ics? It turns out that the two are directly related.
Let us gradually reduce the intensity of light from source 1. The il-
lumination of screen 3 as a result will naturally decrease. However, the
Figure 7.1: The double slit experiment.
interference character of the curve 𝐼 (𝑥) will be retained. By increasing
the time of exposure, it is possible in principle to obtain the interference
curve 𝐼 (𝑥) for even the smallest light intensities. This is not trivial since
with decreasing intensity of the light beam the number of photons in it
will decrease and so, obviously, a situation should arise when individual
photons will have to be considered in place of light waves. However, as
has been shown experimentally, the nature of the interference curve 𝐼 (𝑥)
obviously remains unchanged no matter how much the light intensity
is decreased. The distribution of the individual photons falling on the
detector screen gives the same interference pattern on the screen as is
produced by light waves.
The second fact is connected with the specific nature of the passage
of an electron through the slits in the screen. Let us close slit 𝐵; in this
case we observe the distribution of incidences on the screen, as described
by curve 𝐼 1 (𝑥) (Figure 7.2). Let us open slit 𝐵 but close slit 𝐴; in this case
the distribution 𝐼 2 (𝑥) is observed. By opening both slits, we do not get
the additive distribution 𝐼 1 (𝑥) + 𝐼 2 (𝑥) described by the curve 𝐼 3 (𝑥) in the
diagram but the interference distribution 𝐼 (𝑥) which was noted earlier. It
is this fact that is especially remarkable. If we suppose that each electron Figure 7.2: The double slit experiment,
passes through anyone of the slits, the appearance of the interference with one slit closed.
distribution 𝐼 (𝑥) forces us to admit that the electron in some way “per-
ceives” the other slit, otherwise, it does not matter for an electron passing
through one slit whether the neighbouring slit is open or closed and thus
the distribution of incidences with both slits open must be described not
by the interference curve but by the additive curve 𝐼 3 (𝑥) = 𝐼 1 (𝑥) + 𝐼 2 (𝑥).
The electrons passing through slit 𝐴 should give the distribution 𝐼 1 (𝑥),
while those passing through slit 𝐵 should give the distribution 𝐼 2 (𝑥). The
detector screen should then register the sum of these distributions. Since
it is meaningless to talk about the electron “perceiving” we have to admit
that the interference distribution 𝐼 (𝑥) observed with both the slits open
is associated with the passage of the electron somehow simultaneously
through the two slits.
However, this admission does not simplify matters since it is not clear
72 b a sic co n c e p t s o f q uan t u m m e c h a n ics
Let us imagine that near the slits 𝐴 and 𝐵 of screen 2 we have light Experiment 2 (“Observing” a
sources 4 and photodetectors 5 (Figure 7.3), designed for “observing” the Microparticle in the Interfer-
passage of electron through the screen with slits (the photodetectors reg- ometer)
ister light scattered by the electron). If the electron simultaneously passes
through both the slits both the photodetectors are activated simultane-
ously, But if the electron passes through either one of the slits, only one
detector is activated; in this case we shall also know through which slit
the given electron passes.
We repeat the experiment after switching off the light sources (thus
depriving ourselves of the possibility of “observing” the passage of the
electrons through the slits). In this case we again observe the interference
curve 𝐼 (𝑥).
The situation is such that interference occurs when light sources are
switched off and is absent when they are switched on. As soon as we try
to control the process of passage of electrons through two open slits, the
interference disappears. In other words, our observation of the behaviour
of electrons near the slits destroys the interference!
But, may be, we could think of some other experiment – without re-
sorting to the scattering of photons by electrons? For example, isn’t it
possible to try to construct an extremely light screen with slits in such a
74 b a sic co n c e p t s o f q uan t u m m e c h a n ics
way, that it could move to the left or right upon being struck by individual
electrons? The screen deflects to the left if an electron passes through the
left slit and to the right if it passes through the right slit (Figure 7.4). But
if the electron passes through two slits simultaneously, the screen will
not move at all. Thus, we just have to follow the movement of the screen.
It would appear that at least in principle the aim has been achieved - the
required delicate experiment has been conceived. But it is meaningless
to set up such an experiment. To make sure that this is so, we recall the
uncertainty relation for momentum and coordinate. It follows from this
relation that if the experimental conditions really permit us to register
the momentum of the screen due to a recoil from the electron impinge-
ment, the same condition must lead to an uncertainty in the position of
the screen on the line 𝑂𝑂 (Figure 7.4). Consequently, the shift in such
a screen does not permit one to draw any conclusion on the nature of
passage of an electron through the slits. If, on the other hand, we fix the
position of the screen on the line 𝑂𝑂, then it is easy to see that a measure-
ment of the momentum of its recoil will become impossible.
Figure 7.4: Modifications to the double slit
experiment.
Several attempts were made to devise such an experiment in which
the passage of electrons through a screen with slits could be “controlled”
without seriously influencing the electrons themselves (so that the in-
terference is not destroyed). But all these attempts proved futile. As a
result, we must admit that the above conclusion regarding the destruction
of interference caused by observing the behaviour of electrons near the
slit, is of a fundamental nature. In other words, the effect of observation
(measurement) destroying interference cannot be eliminated in principle.
A Brief Interlude
rea de r: But then experiment 2 did not attain its goal. Was it neces-
sary to consider it?
aut hor: Yes, it was. The experiment did not answer the question
posed by us. So what? It just means that the question was not
ph y s i c al f o u n dat i o n s o f q ua n t u m m e c h a n i c s 75
rea de r: Does the whole idea of the experiment lie in its negative
result?
aut hor: Let us not make haste. We shall first consider our system e
fundamental experiments to the end.
Let us pass a beam of light through a polarizer, say, a tourmaline crystal. Experiment 3 (Passage of
A linearly polarized light beam emerges from the crystal. The direction of Photons Through Polarizers)
polarization of the beam is determined by the orientation of the polarizer
with respect to the beam (the direction of polarization coincides with the
direction of the axis of the polarizer). Let us denote the intensity of the
linearly polarized light beam through 𝐼 .
In case (a) the initial photon always passes through the second po-
larizer; in case; (b), on the contrary, it never passes the second polarizer.
These results are not unexpected. But what happens in case (c)? It turns
out that in this case the photon may pass through the second polarizer or
it may not. Moreover, it is absolutely impossible to predict which of the
two alternatives (passing or not passing) will be realized for a given initial
photon. If it so happens that the photon passes through the second polar-
izer, its polarization will change – it will be polarized in the direction of
the axis of the second polarizer. Thus, the fate of any particular individual
photon is, in principle, unpredictable!
in actual practice it turns out quite differently – now the photons do not
pass through the apparatus at all!
We shall consider elastic collisions of microparticles and use for con- Experiment 4 (Scattering of Mi-
venience the centre of mass system for the colliding particles. Figure 7.7 croparticles by Microparticles)
shows experimental diagram related to the system of the centre of mass
of the particles. Here, 𝐴 and 𝐵 are particle beams, 1 and 2 are the counters
for scattered particles, deployed on the line perpendicular to the direc-
tion of motion of the particles before collision. Thus, we consider here
the scattering of particles through an angle of 90° in the centre of mass
system.
We note that the picture of the process in the centre of mass system
may considerably differ from the analogous picture in the laboratory
system. Thus, for example, in the laboratory system the counters 1 and 2
may not be on the same line. Besides, in actual practice only one beam of
particles (for example, particles of type 𝐴) may be used while the particles
of the other type (type 𝐵) constitute the stationary target. It is assumed
that every time the experiment in the laboratory system is conducted
in such a way that the diagram shown in Figure 7.7 is applicable for the
centre of mass system of the particles.
We shall consider different examples as applied to the above diagram, Figure 7.7: Elastic collision of microparti-
cles.
measuring each time the probability of scattering of particles by the
number of simultaneous activations of counters 1 and 2.
Second example. The particles are the same, but now each counter can
register both 𝛼-particles and 3 He nuclei. In this case the measured
probability of scattering turns out to be 2𝑤. This result appears
quite natural – the doubling of the probability 𝑤 is associated with
the realization of the two alternatives shown in Figure 7.8.
Fourth example. The two electron beams are nonpolarized. Let the
scattering probability measured in this case be 𝑤𝑒 .
Fifth example. The electron beams are polarized but in both directions.
For example, 𝐴-electrons have spin 𝜎 = 1/2 and 𝐵-electrons, 𝜎 =
−1/2. In this case the scattering probability turns out to be 2𝑤𝑒 .
Sixth example. The electron beams are polarized in the same direction.
In this case the counters 1 and 2 are “silent” – the scattering proba-
bility is zero!
of the microparticle between the two given states has, as a rule, a proba-
bilistic character. We therefore introduce into the picture the transition
probability 𝑤𝑠→𝑓 . In quantum mechanics, apart from transition probabil-
ity, the concept of the amplitude of the transition probability h𝑓 | 𝑠i 2 is 2 The treatment of quantum mechanics
on the basis of probability amplitudes is
also introduced. Generally speaking, it is a complex number, the square of given in books by Feynman and Dirac.
whose modulus is equal to the transition probability: Please note that this is not a square of
h𝑓 | 𝑠 i but the footnote number.
Feynman, R. P., Leighton, R. B., and
𝑤𝑠→𝑓 = | h𝑓 | 𝑠i | 2 (8.1)
Sands, M. (1965). Lectures on Physics,
volume 3. Addison-Wesley Reading,
Note that the amplitude of the transition probability is written so that the Mass; Feynman, R. P. and Hibbs, A.
initial state is on the right and the final one on the left (as if it were read (1965). Quantum mechanics and path
integrals. McGraw-Hill, N.Y; and Dirac, P.
from right to left). Henceforth for brevity we shall call the amplitude of A. M. (1958). The Principles of Quantum
the transition probability the transition amplitude (and sometimes even Mechanics. Clarendon Press Oxford
We shall indicate four basic rules of working with transition ampli- Basic Rules of Working with
tudes. These rules should be considered as postulates forming the basis of Amplitudes
a system of quantum-mechanical concept which are in conformity with
experiment.
First Rule. We assume (Figure 8.1 (a)) that there are several physically
indistinguishable ways (paths) in which a microparticle can move
from 𝑠-state to 𝑓 -state. In this case, the resulting transition ampli-
tude is the sum of the amplitudes corresponding to the different
modes of transition:
Õ
h𝑓 | 𝑠i = h𝑓 | 𝑠i𝑖 (8.2)
𝑖
Second Rule. We assume (Figure 8.1 (b))that there are several final states
(𝑓1, 𝑓2, . . . , 𝑓𝑖 , . . .) and that we are considering the probability of
transition to any of these states, no matter which state it is. In this
case, the resulting transition probability | h𝑓 | 𝑠i | 2 is the sum of the
80 bas i c co n c e p ts o f q ua n t u m m e c h a ni cs
Third Rule. Let us assume (Figure 8.1 (c)) that the transition 𝑠 → 𝑓 takes
place through some intermediate state (𝑣-state). In this case we
introduce the idea of the amplitude of the successive transitions
𝑠 → 𝑣 and 𝑣 → 𝑓 (corresponding to the amplitudes h𝑣 | 𝑠i and
h𝑓 | 𝑣i; the resulting amplitude is the product of these amplitudes:
h𝑓 | 𝑠i = h𝑓 | 𝑣i h𝑣 | 𝑠i (8.4)
Fourth Rule. Suppose (Figure 8.1 (d)) have two independent microparti-
cles. Suppose one microparticle undergoes a transition 𝑠 → 𝑓 and
the other simultaneously undergoes a transition 𝑆 → 𝐹 . In this
case the resulting transition amplitude for the system of micropar-
ticles is given by the product of the transition amplitudes for the
individual microparticles:
h𝑓 𝐹 | 𝑠𝑆i = h𝑓 | 𝑠i h𝐹 | 𝑆i (8.5)
We can see that the second, third and fourth rules appear quite natural
since, together with (8.1), they represent well-known theorems, i.e. the
theorem of addition of probabilities (second rule) and the theorem of
Figure 8.1: Rules of working with ampli-
multiplication of probabilities of independent events (third and fourth tudes.
rules). Only the first rule, which may be called the rule of addition of
amplitudes, appears unusual. In a certain sense, the entire system of
quantum-mechanical concepts is based on the rule of the addition of
amplitudes.
Suppose that the transition of a microparticle from the initial to the Distinguishable and Indis-
final state (𝑠 → 𝑓 transition) always takes place through one of the inter- tinguishable Alternatives.
mediate states (𝑣 1, 𝑣 2, . . . 𝑣𝑖 . . .) (Figure 8.1 (e)) this case, one or the other Interference of Amplitudes
mode of transition 𝑠 → 𝑓 (one alternative or the other) is determined by
the “participation” of the corresponding intermediate state in the transi-
tion.
We take two different cases. Suppose in the first case the intermediate
state through which a given transition takes place is known. This is the
case of physically distinguishable alternatives. To describe this we must
ph y s i c al f o u n dat i o n s o f q ua n t u m m e c h a n i c s 81
combine the second and the third rules. The transition probability that we
obtain as a result will be of the form
Õ
| h𝑓 | 𝑠i | 2 = | h𝑓 | 𝑣𝑖 i h𝑣𝑖 | 𝑠i | 2 (8.6)
𝑖
One might ask: where does the second rule come in if it only involves
the various final states? As a matter of fact, if we know the intermediate
state at which the microparticle arrives, it may be treated as the final
state of the first step of the transition. We fix the microparticle in this
state and temporarily stop the experiment here. One microparticle will be
fixed in one state, the others in various other states, so, a situation with
different final states actually arises. We stress here that distinguishability
of alternatives is connected with the actual existence of different final
states (even if in the given experiment they play the role of intermediate
states).
It is interesting to expand somewhat the fourth rule regarding the Transition Involving Two Mi-
simultaneous transition of two microparticles. Let us suppose (Fig- croparticles
ure 8.2 (a)) that one microparticle undergoes a transition 𝑠 → 𝑓 through
the intermediate 𝑣-state and that another microparticle simultaneously
undergoes the transition 𝑆 → 𝐹 through the intermediate 𝑉 -state. By
combining the third and fourth rules we represent the transition ampli-
82 bas i c co n c e p ts o f q ua n t u m m e c h a ni cs
h𝑓 𝐹 | 𝑠𝑆i = h𝑓 | 𝑣i h𝑣 | 𝑠i h𝐹 | 𝑉 i h𝑉 | 𝑆i (8.9)
Let us assume further (Figure 8.2 (b)) that both the microparticles in
the process of their transitions ass through one and the same intermediate
𝑣 1 -state. Then (8.9) must take the form
h𝑓 𝐹 | 𝑠𝑆i = h𝑓 | 𝑣 1 i h𝑣 1 | 𝑠i h𝐹 | 𝑣 1 i h𝑣 1 | 𝑆i (8.10)
Finally, we assume (Figure 8.2 (c)) that each microparticle realizes a num-
ber of physically indistinguishable alternatives through different interme-
diate states (𝑣 1, 𝑣 2, . . . , 𝑣𝑖 ). Moreover, every intermediate state is common
to both the microparticles. In this case generalizing result (8.10) by com-
bining it with the first rule, we get
Õ
h𝑓 𝐹 | 𝑠𝑆i = h𝑓 | 𝑣𝑖 i h𝑣𝑖 | 𝑠i h𝐹 | 𝑣𝑖 i h𝑣𝑖 | 𝑆i (8.11)
𝑖
Since the 𝐹𝑖 -states are the various final states, we get, according to second
rule, the following expression for the resulting transition probability of
the “controlled” 𝑠-particle:
Õ
| h𝑓 | 𝑠i | 2 = | h𝑓 𝐹𝑖 | 𝑠𝑆i | 2
𝑖
Õ (8.13)
= | h𝑓 | 𝑣𝑖 i h𝑣𝑖 | 𝑠i h𝐹𝑖 | 𝑣𝑖 i h𝑣𝑖 | 𝑆i | 2
𝑖
If we further assume that the amplitude h𝐹𝑖 | 𝑆i is the same for all 𝑖
(which is often the case in practice), then, denoting this amplitude by 𝑎
for brevity, we rewrite (8.13) in the form
Õ
| h𝑓 | 𝑠i | 2 = | 𝑎 | 2 | h𝑓 | 𝑣𝑖 i h𝑣𝑖 | 𝑠i | 2 (8.14)
𝑖
Thus while result (8.8) is obtained in the absence of 𝑆-particles (in the
absence of “control”), we now have the result (8.14). It is easy to see that
it corresponds to (8.6) - we destroy the interference of amplitudes by
establishing a “control” over the intermediate states, i.e. by turning the
physically indistinguishable alternatives into distinguishable ones.
Using the concept of the transition amplitude and the rules relating Behaviour of a Microparticle
to the amplitudes, let us turn to Experiment 1 discussed in Section 7. An in the Interferometer and
electron emerges from the initial 𝑠-state, passes through a screen with Interference of Amplitudes
slits 𝐴 and 𝐵, each of which corresponds to its intermediate state (𝐴-state
and 𝐵-state, respectively) and is finally registered in its final 𝑥-state, i.e.
at the point with coordinate 𝑥 on the detector screen. Suppose that slit
𝐴 is open and slit 𝐵 is closed. In this case h𝑥 | 𝑠i𝐴 = h𝑥 | 𝐴i h𝐴 | 𝑠i. The
probability of transition 𝑠 → 𝑥, i.e. the probability of the electron being
registered at point 𝑥 of the detector screen, is of the form
| h𝑥 | 𝑠i𝐴 | 2 = | h𝑥 | 𝐴i h𝐴 | 𝑠i | 2 (9.1)
We denote this probability by 𝐼 1 (𝑥), and recall that this is how we denoted
the distribution of electron incidences on the detector screen in Exper-
iment 1 (Section 7) under the conditions that slit 𝐴 is open and slit 𝐵 is
closed. For the probability of an electron being registered at point 𝑥 in the
case when slit 𝐵 is open and slit 𝐴 is closed, we may write the analogous
expression
| h𝑥 | 𝑠i 𝐵 | 2 = | h𝑥 | 𝐵i h𝐵 | 𝑠i | 2 = 𝐼 2 (𝑥) (9.2)
84 bas i c co n c e p ts o f q ua n t u m m e c h a ni cs
Now let us open both the slits. Since it is impossible to indicate through
which slit any electron passes (the alternatives are indistinguishable), we
have, consequently,
h𝑥 | 𝑠i = h𝑥 | 𝐴i h𝐴 | 𝑠i + h𝑥 | 𝐵i h𝐵 | 𝑠i (9.3)
𝐼 (𝑥) = | h𝑥 | 𝐴i + h𝐴 | 𝑠i + h𝑥 | 𝐵i h𝐵 | 𝑠i | 2
(9.4)
= 𝐼 1 (𝑥) + 𝐼 2 (𝑥) + h𝑥 | 𝑠i𝐴 h𝑥 | 𝑠i 𝐵∗ + h𝑥 | 𝑠i𝐴∗ h𝑥 | 𝑠i 𝐵
Figure 9.1 depicts schematically the fundamental Experiment 2 consid- Destruction of Interference of
ered in Section 7. Here 𝑠 is the electron source, 𝑆 is the photon source, 𝐹 1 Amplitudes upon “Controlling”
and 𝐹 2 are photoelectric counters which fix the two final states of photons the Behaviour of a Microparti-
scattered by electrons in the vicinity of slits 𝐴 and 𝐵. cle in the Interferometer
To begin with, we shall assume that the photons scattered in the vicin-
ity of either of the slits may be registered in the 𝐹 1 -state as well as the
𝐹 2 -state (which corresponds to the use of radiation with a sufficiently
large wavelength). In this case, obviously, the photons don’t “control”
the passage of electrons through the screen with slits. We denote the
transition amplitudes thus:
for electrons
h𝑥 | 𝐴i h𝐴 | 𝑠i = 𝜑 1
(9.5)
h𝑥 | 𝐵i h𝐵 | 𝑠i = 𝜑 2
for photons (taking into account the symmetry of the photon transition
h𝐹 1 | 𝐴i h𝐴 | 𝑆i = h𝐹 2 | 𝐵i h𝐵 | 𝑆i = 𝜓 1
(9.6)
h𝐹 2 | 𝐴i h𝐴 | 𝑆i = h𝐹 1 | 𝐵i h𝐵 | 𝑆i = 𝜓 2
Using these notations and result (8.11), we write the following ex-
pression for the probability amplitude of simultaneously registering an
electron at point 𝑥 and a photon in the 𝐹 1 -state
| h𝑥 | 𝑠i | 2 = (| 𝜑 1 | 2 + | 𝜑 2 | 2 )(| 𝜓 1 | 2 + | 𝜓 2 | 2 )
(9.10)
+ (𝜑 1𝜑 2∗ + 𝜑 1∗𝜑 2 ) (𝜓 1𝜓 2∗ + 𝜓 1∗𝜓 2 )
| h𝑥 | 𝑠i | 2 = | 𝜓 1 | 2 (| 𝜑 1 | 2 + | 𝜑 2 | 2 ) (9.11)
From this example we see that there is a subtle point involved in the
question of the distinguishability of alternatives: in addition to the com-
plete indistinguishability and complete distinguishability, there is a con-
tinuous spectrum of intermediate situations which should be identified
with partial distinguishability. The result (9.11) describes the limiting case
of complete indistinguishability of the alternatives under consideration
(𝜓 2 = 0). The opposite extreme case of the complete indistinguishability
of alternatives envisages equal probabilities for a photon falling on its
“own” or the “other” detector: 𝜓 1 = 𝜓 2 . In this case it is easy to see that
expression (9.10) assumes the form
| h𝑥 | 𝑠i | 2 = 2| 𝜓 1 | 2 | 𝜑 12 + 𝜑 22 | 2 (9.12)
Results (9.11) (the squares of the moduli of electron amplitudes are added)
and (9.12) (the electron amplitudes themselves are added up) are obtained
from (9.10) as particular (limiting) cases. The general expression (9.10)
describes the intermediate situation corresponding to partial distinguisha-
bility of the alternatives under consideration, differing from one another
by the magnitude of the interference component. The less the interference
component is the greater is the degree of distinguishability of alternatives.
We turn to Experiment 4 of Section 7. Let 𝑠 1 and 𝑠 2 be the initial states Scattering of Microparticles
of the colliding microparticles and 𝑓1 and 𝑓2 be the final states registered and Interference of Amplitudes
by the corresponding counters. In Section 7 we considered the scattering
ph y s i c al f o u n dat i o n s o f q ua n t u m m e c h a n i c s 87
through an angle of 90° in the centre of mass system of the colliding par-
ticles. For a more general approach, we shall consider scattering through
an angle 𝜃 . In this case the counters are arranged along a straight line
at an angle 𝜃 with the initial direction of the colliding particles - see Fig-
ure 9.2 (a) (the analysis is carried out, as before, in the centre of mass
system of particles).
ternatives shown in Figure 9.2 (b), (c). In this case we should sum not
the probabilities of the alternatives, but their amplitudes. The probability
of simultaneous activation of the counters should be determined by the
expression
𝑤 = | 𝜑 (𝜃 ) | + 𝜑 (𝜋 − 𝜃 ) | 2 (9.17)
𝑤 = | 𝜑 (𝜋/2) + 𝜑 (𝜋/2) | 2
(9.18)
= 4 | 𝜑 (𝜋/2) | 2
𝑤 = | 𝜑 (𝜃 ) ± 𝜑 (𝜋 − 𝜃 ) | 2 (9.19)
The alternative with the “+” sign (interfering amplitudes have the same
sign) is already familiar to us - it is the expression (9.17). The other alter-
native, when the amplitude with opposite signs interfere, is also formally
possible. It is remarkable that nature “employs” this alternative as well;
this can be verified by studying the results of experiments on scattering of
electrons by electrons.
𝑤𝑒 = | 𝜑 (𝜃 ) − 𝜑 (𝜋 − 𝜃 ) | 2 (9.20)
and turn to the results of the indicated experiments. For 𝜃 = 𝜋/2, the
probability (9.20) vanishes. This corresponds to the sixth example in
Experiment 4 of Section 7. We recall that this example concerned the
collision of electrons in the same spin state. It is in this case that we have
two completely indistinguishable alternatives for the electrons.5 5 We shall henceforth omit the word
“completely”, but shall always mean it
while speaking of distinguishable and
If the colliding electrons are in different spin states (the fifth example indistinguishable alternatives. Partial
in Experiment 4), the alternatives are distinguishable. In this case the distinguishability will be specially men-
tioned.
probability of the activation of the counters is given (as for distinguish-
able particles) by expression (9.17), which for 𝜃 = 𝜋/2 leads to the familiar
result 2 | 𝜑 (𝜋/2) | 2 In the case of non-polarizedelectron beams (the fourth
example in Experiment 4), it should be remembered that the probability
ph y s i c al f o u n dat i o n s o f q ua n t u m m e c h a n i c s 89
1
of collision between two electrons in similar spin states is . From this,
2
taking (9.20) and (9.16) into account, we get the following expression for
the probability of activation of the counters
1
𝑤𝑒 = | 𝜑 (𝜃 ) − 𝜑 (𝜋 − 𝜃 ) | 2
2 (9.21)
1
+ | 𝜑 (𝜃 ) | 2 + | 𝜑 (𝜋 − 𝜃 ) | 2
2
Result (9.21) includes the summing of amplitudes (for cases characterized
by indistinguishable alternatives) as well as the summing of probabilities
(for cases characterized by distinguishable alternatives). For 𝜃 = 𝜋/2,
(9.21) gives the probability 𝑤𝑒 = | 𝜑 (𝜋/2) | 2 .This is half the “classical prob-
ability” (i.e. the probability taking place in the case of indistinguishable
alternatives) in complete agreement with the results of the experiments
considered in Section 7.
Thus, we have found that the experiments on the scattering of mi- Interference of Amplitudes and
croparticles described in Section 7 provide a good experimental back- Division of Microparticles into
ground for the concept of the interference of amplitudes. Moreover, these Bosons and Fermions
experiments indicate the necessity for using not one but two different
laws of interference, (9.17) and (9.20). We shall discuss the meaning of
these two laws, assuming that 𝜃 = 𝜋/2. According to (9.17), we have for
𝛼-particles
𝑤 = 4 | 𝜑 (𝜋/2) | 2 (9.22)
and from (9.20) we have for the electrons in the same spin state
𝑤𝑒 = 0 (9.23)
The fact that all microparticles in nature are divided, according to their
behaviour in assemblies of similar particles, into two groups – bosons
(with a tendency to densely “populate” the same state) and fermions (“pop-
ulating” the states only one at a time) has already been mentioned in
Section 1. Now we see that this fundamental fact is associated with the
existence of two different laws for the interference of amplitudes. In the
case of bosons, the amplitudes with like signs interfere; in the case of
90 bas i c co n c e p ts o f q ua n t u m m e c h a ni cs
Let us consider an example: there are three atoms emitting photons Bosonic Nature of Photons
(𝑠 1, 𝑠 2, 𝑠 3 ) and three photon counters (𝑓1, 𝑓2, 𝑓3 ). The amplitude of prob- and Processes of Spontaneous
ability that three transitions 𝑠 1 → 𝑓1, 𝑠 2 → 𝑓2, 𝑠 3 → 𝑓3 take place and Induced Emission of Light
simultaneously is h𝑓1 | 𝑠 1 i h𝑓2 | 𝑠 2 i h𝑓3 | 𝑠 3 i. We assume that the photons are
registered in the same state. We then have 3! = 6 indistinguishable alter-
natives. Besides the one indicated above, the remaining five amplitudes
are given by
𝑤𝑛 = 𝑛!𝑊𝑛 (9.24)
𝑤𝑛+1 𝑊𝑛+1
= (𝑛 + 1) (9.26)
𝑤𝑛 𝑊𝑛
This means that the probability of getting one more boson in a state
where there are already 𝑛 bosons is (𝑛 + 1) times greater than the proba-
bility of getting one more distinguishable microparticle in a state where
there are already 𝑛 such microparticles. We note further that for distin-
guishable microparticles the degree of prior “population” of the state is
not important. When applied to bosons, it is analogous to the situation
ph y s i c al f o u n dat i o n s o f q ua n t u m m e c h a n i c s 91
when a boson appears in a state which was not occupied earlier. Hence
result (9.26) may also be interpreted in a different way. The probability
of getting another boson in a state having 𝑛 bosons is 𝑛 + 1 times greater
than the probability of a boson appearing in a state which was hitherto
unoccupied. In accordance with this interpretation we can rewrite result
(9.26) in the form
| h𝑛 + 1 | 𝑛i | 2 = (𝑛 + 1)| h1 | 0i | 2 (9.27)
| h𝑛 + 1 | 𝑠i |𝑠2 = | h1 | 0i | 2 (9.29)
| h𝑛 + 1 | 𝑛i |𝑖2 = 𝑛 | h1 | 0i | 2
(9.30)
Experiment shows that the probability of the absorption of light by a Absorption of Light and Con-
substance depends upon the number of photons in the radiation. In this nection Between the Ampli-
respect the process of absorption of light is an induced process. By using tudes of Direct and Reverse
the analogy with (9.30) we can write the following expression for the Transitions
probability of the annihilation of a photon in a state having 𝑛 photons:
| h𝑛 − 1 | 𝑛i | 2 = 𝑛 | h0 | 1i | 2 .
92 bas i c co n c e p ts o f q ua n t u m m e c h a ni cs
| h𝑛 | 𝑛 + 1i | 2 = (𝑛 + 1) | h0 | 1i | 2 (9.31)
| h𝑛 + 1 | 𝑛i | 2 = | h𝑛 | 𝑛 + 1i | 2 .
| h𝑓 | 𝑠i | 2 = | h𝑠 | 𝑓 i | 2 (9.32)
h𝑓 | 𝑠i = h𝑠 | 𝑓 i ∗ (9.33)
We shall consider two useful examples which demonstrate the am- Supplementary Examples
plitude concepts very clearly. Scattering of neutrons by a crystal. We
shall consider the scattering of very slow neutrons (with energies of the
order of 0.1 eV and lower) by atomic nuclei. It is well known that for such
low energies, the scattering amplitude 𝜑, considered in the centre of mass
system for a neutron and a nucleus, is independent of the scattering angle.
So, it would appear, the scattering probability should also be isotropic.
However, experiments on the scattering of very slow neutrons by crys-
tals reveal a strong angular dependence of the scattering probability. A
typical curve is shown in Figure 9.3: peaks are observed against a smooth
background. These peaks are a visual demonstration of the effect of the
interference of amplitudes. We shall see how this is so.
h𝑓 | 𝑠i𝑖 = h𝑓 | 𝑖 i 𝜑 h 𝑖 | 𝑠i (9.34)
We assume that all the 𝑁 nuclei of the crystal are alike, have zero spin, Figure 9.3: Results of scattering of neu-
trons by a crystal.
and are located strictly at the lattice points.
Õ
𝑁
h𝑓 | 𝑠i = h𝑓 | 𝑠i𝑖
𝑖
(9.35)
Õ
𝑁
= h𝑓 | 𝑖 i 𝜑 h 𝑖 | 𝑠i
𝑖
Taking (9.35) into account, the probability of scattering of the neutron has
the form 2
Õ𝑁
| h𝑓 | 𝑠i | 2 = | 𝜑 | 2 h𝑓 | 𝑖 i h 𝑖 | 𝑠i (9.36)
𝑖
We further assume that nuclei in the crystal have a spin (let it be equal
to 1/2, as for neutrons). In this case we should distinguish between the
amplitude of scattering by the nucleus with spin inversion (in accordance
with the law of conservation of momentum, the neutron spin must also he
inverted in this case), and without spin inversion of the nucleus (and the
neutron), i.e. between 𝜒 and 𝜑, respectively. If the scattering of the collid-
ing microparticles is accompanied by spin inversion, the corresponding
alternative is distinguishable - it is clear that in such an act of scattering
only that nucleus participates whose spin has been inverted. Now the
probabilities and not the amplitudes should be summed.
The first term on the right-hand side of (9.37) accounts for the character-
istic interference peaks in Figure 9.3, while the second term is responsible
for the smooth background. It is customary to say that the first term de-
scribes the probability of the coherent scattering, and the second that of
the incoherent scattering of neutrons.
(a) observations are carried out to find the state of the atoms in the beam
between fields 𝐵 1 and 𝐵 2 , and when
In case (b) the intermediate 𝑖-states are not fixed and, consequently, the
alternatives corresponding to them are indistinguishable. Hence
2 Figure 9.4: Path integral approach.
Õ
| h𝑓 | 𝑠i | =
2
h𝑓 | 𝑖i h𝑖 | 𝑠i (9.39)
𝑖
The concept of the motion of a microparticle along classical path integrals (in other words, through
interference of amplitudes corresponding to the classical trajectories) is discussed in detail in Section 5.
ph y s i c al f o u n dat i o n s o f q ua n t u m m e c h a n i c s 95
10 Superposition of States
Earlier, in Section 3, we studied the uncertainty relations. In this con- Principle of Superposition of
nection it was remarked, in particular, that the states of a microparticle States
were combined in groups each of which contained the definite values of
anyone complete set of physical quantities. We also gave examples of
complete sets of values for an electron and a photon.
The reader must be warned at once that the above impression is erro-
neous. However, in order to be convinced of this, we must analyse the
process of measurement. Measurement in quantum mechanics will be
discussed in Section 11. This discussion will necessitate some correction
in the above definition of the amplitudes h𝛼 | 𝛽i and will enable us to actu-
ally reduce the amplitudes of states to the already familiar amplitudes of
transitions.
But everything has its own time, and so, for the time being, we shall
operate with the concept of “amplitude of state” as an independent con-
cept, without ascertaining the practical meaning of, say, the phrase “a
particle in state h𝛼 | can also be found in state h𝛽 |”.
ing, the uncertainty relations indicate the “old” concept which must be
rejected while we go over from macrophenomena to microphenomena. In
particular, they require a rejection of the simultaneous measurability of
all physical quantities characterizing a given object. At the same time, the
principle of superposition indicates the “new” concept which is applica-
ble when considering microparticles; superposition (10.2) means that if a
microparticle is in a state in which the quantities of the 𝛼-set are measur-
able, then the value of the quantities in the 𝛽-set may be predicted with a
probability equal to | h𝛼 | 𝛽i | 2 .
In classical physics one comes across superposition quite frequently, Superposition in Classical
the foremost example being the well-known superposition of classical Physics and Quantum Me-
waves. From a mathematical point of view, the classical superposition and chanics
superposition in quantum mechanics are analogous. This circumstance
greatly stimulated the development of quantum theory. At the same
time, it certainly complicated the interpretation of the physical content
of theoretically obtained results since it tempted one to draw erroneous
analogies with classical waves. In the words of Dirac6 : 6 Dirac, P. A. M. (1958). The Principles
of Quantum Mechanics. Clarendon Press
Oxford
the assumption of superposition relationships between the states leads to a
mathematical theory in which the equations that define a state are linear
in the unknowns. In consequence of this, people have tried to establish
analogies with systems in classical mechanics, such as vibrating strings or
membranes, which are governed by linear equations and for which, there-
fore, a superposition principle holds . . .(remember the criticism in Section 5
of the attempts to represent the motion of a bound microparticle with the
help of classical waves in a resonator - author’s remarks). It is important
to remember, however, that the superposition that occurs in quantum
mechanics is of an essentially different nature from any occurring in the
classical theory, as is shown by the fact that the quantum superposition
principle demands indeterminacy in the results of observations.
h𝛼 | 𝛽 1 i h𝛽 1 | + h𝛼 | 𝛽 2 i h𝛽 2 |
In order to finally convince ourselves that the quantum-mechanical Mutually Orthogonal States:
principle of superposition has in fact nothing in common with the clas- Total and Partial Distinguisha-
sical superposition we turn to expression (10.2) and see how it changes bility of States
upon the transition to classical physics. Since in classical physics all the
quantities can be simultaneously measured, they form together one “com-
plete set”. Considering that the superposition bonds described by relation
(10.2) operate between different complete sets, we arrive at the conclusion
that in the classical case such bonds simply do not exist and, consequently,
all formally composed amplitudes of states must be taken as being equal
to zero:
h𝛼 | 𝛽i = 0 (10.4)
In quantum mechanics condition (10.4) also exists, but only “within the
limits” of the given complete set (for states belonging to the same set).
Thus, for example,
𝛼𝑖 𝛼 𝑗 = 0, if 𝑖 ≠ 𝑗 (10.5)
The amplitude of a state is equal to zero if and only if the two correspond-
ing states are mutually independent (if an object is in one of these states, it
cannot be found in the other). Such states are called mutually orthogonal.
ph y s i c al f o u n dat i o n s o f q ua n t u m m e c h a n i c s 99
In this respect all the states of a classical particle are mutually orthogo-
nal, while in quantum mechanics only the states belonging to the same
complete set are orthogonal and the states belonging to different sets are
non-orthogonal. This last fact is reflected in the principle of superposition
of states.
h𝑠 1 | = h𝑠 1 | 𝛼 1 i h𝛼 1 | + h𝑠 1 | 𝛼 2 i h𝛼 2 |
(10.6)
h𝑠 2 | = h𝑠 2 | 𝛼 1 i h𝛼 1 | + h𝑠 2 | 𝛼 2 i h𝛼 2 |
It can be shown7 that in this case the probability of simultaneous activa- 7 Helfer, Y. M., Lyuboshitz, V. L., and
Podgoretsky, M. I. (1975). Gibbs’ Paradox
tion of detectors is determined by the expression and Identity in Quantum Mechanics (in
Russian). Nauka, Moscow
𝑤 = | 𝜑 (𝜃 ) | 2 + | 𝜑 (𝜋 − 𝜃 ) | 2
(10.7)
+ | h𝑠 1 | 𝑠 2 i | 2 [𝜑 (𝜃 ) 𝜑 ∗ (𝜋 − 𝜃 ) + 𝜑 ∗ (𝜃 ) 𝜑 (𝜋 − 𝜃 )]
For | h𝑠 1 | 𝑠 2 i | 2 → 0, the result (10.7) turns into (9.16) (we come to the
limiting case of complete distinguishability). For | h𝑠 1 | 𝑠 2 i | 2 → 1, the
result (10.7) turns into (9.17) (we come to the limiting case of complete
indistinguishability).
100 b as ic c on c e pt s o f q ua n t u m m e ch a n ics
Thus, we find that the question of the complete and partial distin-
guishability of alternatives in quantum mechanics is closely linked with
the quantum-mechanical principle of superposition, more precisely, with
the mutual orthogonality or non-orthogonality of states.
The different states corresponding to the same complete set of quan- Basic States
tities are called basic states (eigenstates). The amplitudes of elementary
states satisfy the condition
𝑎𝑖 𝑎 𝑗 = 𝛿 𝑖 𝑗 , (10.8)
The principle of superposition permits an explanation of the results of Superposition of States and
experiment 3 in Section 7. Using this principle, we shall consider the Passage of Photons Through
passage of individual photons through the system of three polarizers Polarizers
shown in Figure 7.6. We denote the state of polarization of a photon after
the first polarization by h𝑠 |. According to the principle of superposition,
the state h𝑠 | may be considered as a superposition of the basic states h1|
andh2|, corresponding to two independent polarizations of the photon-
along and perpendicular to the axis of the second polarizer, respectively:
h𝑠 | = h𝑠 | 1i h1| + h𝑠 | 2i h2 | (10.10)
(note that in this example, the system of basic states contains only two
states). The amplitudes of the states may be written in this case in the
form h𝑠 | 1i = cos 𝛼 and h𝑠 | 2i = sin 𝛼. Thus,
The second polarizer lets through photons from the state h1 | only. Since
according to (10.11) the state h1 | is “represented” in state h𝑠 | with a prob-
ability cos2 𝛼, out of 𝑁 photons only 𝑁 cos2 𝛼: photons will pass through
the second polarizer. Moreover, all the photons that pass will appear in
the state h1 | (i.e. they will be polarized long the axis of the second po-
larizer). Thus, in front of the second polarizer, the photon exists as if
partially in the state h1 | and partially in the state h2 |. At the instant when
the photon passes through the polarizer, this “duality” vanishes. In some
cases a photon exists in the state h2 | and so cannot pass through the po-
larizer, while in some other cases it is in the state h1 | and can thus pass
through the polarizer. Further, for any individual photon it is impossible
to predict in which state it will appear (hence it is impossible to predict
whether a given photon will pass through the polarizer or not).
The third polarizer lets through photons in the state h10 | only. This state
is “represented” in the state h10 | with a probability sin2 𝛼. Hence out of
𝑁 cos2 𝛼, photons only 𝑁 cos2 𝛼 sin2 𝛼 photons will pass through the third
polarizer, all these photons being in the state h10 |. If we now remove the
second polarizer, then in place of (10.11) and (10.12) we get
Let the transition from state h𝑠 | to state h𝑓 | take place through certain Principle of Superposition
intermediate 𝑣-states. We suppose that the microparticle is not fixed of States and Interference of
in the intermediate state so that a case of physically indistinguishable Transition Amplitudes
alternative takes place. In this case, as we know, the transition amplitude
h𝑓 | 𝑠i given by the expression
Õ
h𝑓 | 𝑠i = h𝑓 | 𝑣𝑖 i h𝑣𝑖 | 𝑠i (10.14)
𝑖
h𝑓 | 𝑣𝑖 i = h𝑓 | 𝑣𝑖 i (10.17)
We shall demonstrate the basic methods reflecting, in Feynman’s The Mechanics of Quantum
words, the “mechanics of quantum mechanics”. Let us delete the state h𝑓 | Mechanics
from the left- and right- hand sides of equation (10.14). This gives
Õ
h𝑠 | = h𝑣𝑖 | = h𝑣𝑖 | 𝑠i (10.18)
𝑖
Let us further assume that some apparatus converts state h𝑠 | into some
other state h𝑠 0 |. We express this in the general form as
𝐴 | 𝑠 i = | 𝑠 0i (10.19)
ph y si c al f o u n dat i o n s o f q ua n t u m m e c h a n i c s 103
We shall say that the operator 𝐴 has acted on the state | 𝑠 i giving rise to
state | 𝑠 0i. We shall apply the operator 𝐴 to both sides of equation (10.18).
Using (10.19), we get
Õ
| 𝑠 0i = 𝐴 | 𝑣𝑖 i h𝑣𝑖 | 𝑠 i (10.20)
𝑖
h𝑓 | 𝑠i = h𝑓 | 𝑠i
Õ
h𝑓 | 𝑠i = h𝑓 | 𝑣𝑖 i h𝑣𝑖 | 𝑠i
𝑖
Õ
| 𝑠i = h𝑣𝑖 | h𝑣𝑖 | 𝑠i
𝑖
Õ
𝐴 | 𝑠i = 𝐴 h𝑣𝑖 | h𝑣𝑖 | 𝑠i
𝑖
Õ
𝑣𝑗 𝐴 𝑠 = 𝑣 𝑗 𝐴 𝑣𝑖 h𝑣𝑖 | 𝑠i
𝑖
Õ
ÕÕ
𝑓 𝑣𝑗 𝑣𝑗 𝐴 𝑠 = 𝑓 𝑣 𝑗 𝑣 𝑗 𝐴 𝑣𝑖 h𝑣𝑖 | 𝑠i
𝑗 𝑗 𝑖
ÕÕ
h𝑓 | 𝐴 | 𝑠i = 𝑓 𝑣 𝑗 𝑣 𝑗 𝐴 𝑣𝑖 h𝑣𝑖 | 𝑠i
𝑗 𝑖
Finally, we assume that operator 𝐴 acts on the state | 𝑠i, and is followed
by operator 𝐵. If the reader has mastered the logic of the “mechanics
of quantum mechanics” (it would have been more accurate to call it the
“algebra of quantum mechanics”), he will at once surmise that
ÕÕÕ
h𝑓 | 𝐵𝐴 | 𝑠i = h𝑓 | 𝑣𝑘 i 𝑣𝑘 𝐵 𝑣 𝑗 𝑣 𝑗 𝐴 𝑣𝑖 h𝑣𝑖 | 𝑠i (10.24)
𝑘 𝑗 𝑖
104 b as ic c on c e pt s o f q ua n t u m m e ch a n ics
Suppose that a microparticle is in a certain state h𝛼 |. According to the The Origin of the Superposi-
principle of superposition, the state h𝛼 | may be expanded in terms of any tion of States and the Meaning
system of basic states, for example, in the {h 𝛽𝑖 |} system: of the Amplitudes of States
Õ
h𝛼 | = h𝛼 | 𝛽𝑖 i h 𝛽𝑖 | (11.1)
𝑖
The reader who has read the previous section is already familiar with
all this. It is appropriate now to make things more precise.
Third question: It was agreed earlier (see Section 8) to read the tran-
sition amplitudes from right to left. If h𝛼 | 𝛽𝑖 i is also transition
amplitude, it should also be read in the reverse direction (from left
to right). Isn’t it confusing? Actually, if we strictly follow the con-
dition of writing the preceding states to the right of the ones that
follow, then (11.1) should be written as
Õ
| 𝛽𝑖 i h 𝛽𝑖 | 𝛼 i = | 𝛼 i (11.1a)
𝑖
106 b as ic c on c e pt s o f q ua n t u m m e ch a n ics
The reader has in fact already come across analyzers each time the Examples of Analyzers
interference of transition amplitudes was considered. We shall give a few
examples.
First example: [see (9.3)] - the analyzer is a screen with two slits. It
gives rise to the superposition
h𝑠 | = h𝑠 | 𝐴 i h𝐴 | + h𝑠 | 𝐵 i h𝐵 | (11.2)
The process of measurement in quantum mechanics consists of three The Essence of Measuring
successive stages: Process
Here the arrow 1 corresponds to the working stage, and the arrow 2 to the
registering stage.
In the first place we note that the process of measurement has a radical Some Peculiarities of the
influence on a microparticle. It is enough to point out that a change in Quantum-Mechanical Mea-
the initial state of the microparticle in the measuring process is a circum- suring Process
stance of fundamental importance. It is well known that while carrying
out measurements with macroscopic bodies it is possible to isolate the
object to a certain extent from the means of measurement. In quantum
mechanics this is in principle impossible to do so. In other words, it is
impossible to neglect the interaction of the microparticle with its sur-
roundings.
the microparticle into the two states comprising the superposition). Detec-
tors are placed in the path of each of the wave packets. It is known that
each time only one detector is activated. Suppose that at a certain instant
the detector placed in the path of the reflected part of the wave packet is
activated. This means that the other part of the wave packet momentarily
disappears from that part of the space where the unactivated detector is
placed, and reappears in front of the second detector the moment before
the act of registration. The absurdity of such “behaviour” of the micropar-
ticle, which, by the way, “cannot know” which detector is activated in a
given case, is quite obvious.
When discussing the idea of quantum-mechanical principle of superpo- Potentialities and Their Re-
sition we come to a situation which is analogous to the one encountered alization in the Measuring
when discussing the idea of wave-particle duality (see Section 5). In both Process
cases, a visual (classical) interpretation is not possible. In both cases we
come to a problem connected with potentialities and with their realiza-
tion.
The possibility and the actuality are the well-known categories of mate-
rialistic dialectics. The contradiction that exists between them disappears
every time a possibility is realized in one way or the other. Every particu-
lar situation is characterized by a set of possibilities out of which only one
is realized. The realization process is irreversible; as soon as it is accom-
plished, the initial situation qualitatively changes (one of the possibilities
is realized at the expense of all the other possibilities). The possibility that
has been realized corresponds to a new situation with new possibilities.
The process of resolving the contradictions between the possible and the
actual thus turns out to be endless.
Goethe (Faust)
rea de r: I would like to return to the question of classical superpo- Participants: the Author and
sition. Sometimes I get a seditious idea: what is wrong with such a the Reader.
superposition? In any case it explains the interference effect more
clearly than the superposition of amplitudes. I have due regard for
the structure of quantum-mechanical concepts based on working
with probability amplitudes, but still this idea keeps on haunting me
from time to time.
rea de r: And what about wave quantities such as the electron wave
vector or its wavelength? They also appear in mathematical expres-
sions.
rea de r: But we also have the classical light waves, or, in other
words, photon waves. Aren’t the wave vector and the wavelength
the parameters of a wave in this case also?
aut hor: You are right. By the way, that is the reason why the in-
terference of light was discovered long before the interference of
electrons.
rea de r: It turns out that there are two kinds of interference phe-
nomena in nature; classical interference resulting from a summation
of waves, and quantum-mechanical interference where the probabil-
ity amplitudes are summed.
rea de r: I have already pointed out that so far, you have said noth-
ing about the wave function, although it is widely used in all books
on quantum mechanics.
aut hor: For the same reason we shall also introduce the wave func-
tion at a later stage. In fact we have already introduced it since the
“wave function” and “probability amplitude” are terms describing
the same thing. However, when speaking of the physical founda-
tions of quantum mechanics, it is better to use the term “probability
amplitude”. We can go over to the wave function when we consider
the mathematical apparatus of the theory.
ph y si c al f o u n dat i o n s o f q ua n t u m m e c h a n i c s 115
In conclusion I would like to stress that it is not the terms that are
important, but the way they are used. We can use the term “wave”
for a microparticle, but we must not forget its specific nature. Here
it is worthwhile recalling the remark made in Section 5 about the
impropriety of using the model of a classical wave in a resonator for
a bound electron.
In quantum mechanics the principle of causality refers to the possibil- The Specific Nature of the
ities of the realization of events (properties). In other words, in quantum Quantum-Mechanical Concept
mechanics it is not individually realized events that are causally related, of Causality
but only the possibilities of the realization of these events. This is the
essence of the quantum-mechanical meaning of causality. As Pauli stated
in his Nobel lecture,
Since possible and actually realized events are identical in classical me-
chanics, it is clear that upon a transition from the quantum to the classical
description of the world, the causal relation between the possible events
must be converted into causal relation between realized events. In this
sense the quantum-mechanical principle of causality is a generalization of
the principle of classical determinism - it turns into the latter when going
over from microphenomena to macrophenomena.
One may ask: If in quantum mechanics it is not the realized events but The Manifestation of Causality
in Microphenomena
116 b as ic c on c e pt s o f q ua n t u m m e ch a n ics
rather the possibilities of their realization that are causally related, then
how can the observer make use of such a causality? In an experiment one
always has to deal with events that have been realized.
1 Õ
𝑁
h𝛽 i = 𝛽𝑖 (12.1)
𝑁 𝑖
Suppose that the observer decides to repeat his observations the next day
(or the next year). He will get a certain set of values 𝛽 10 , 𝛽 20 , 𝛽 30 , . . .. The
new set of values will be different from the old set, yet the new mean
value determined by a formula of the type (12.1) will be close to the mean
value h 𝛽 i obtained earlier (provided, of course, that 𝑁 is sufficiently
large). This means that there was no need for the observer to toil on
the second day. The mean value could be predicted on the basis of the
previous day’s measurements.
We assume that the reader has already understood the crux of the prob-
lem. This lies in the fact that the causal relationship among possible
events signifies a causal relationship among the probabilities of the real-
ization of these events. In short, prediction in quantum mechanics has
a probability character! In order to predict the quantity h 𝛽 i in the state
h 𝛼 |, we must know the probability | h 𝛼 |𝛽𝑖 i | 2 of the realization of values
ph y si c al f o u n dat i o n s o f q ua n t u m m e c h a n i c s 117
𝛽𝑖 in the given state. But if these probabilities are not known beforehand
one must collect the corresponding statistics of the measurements of
𝛽-quantities which allow us to find the required probability.
It has already been remarked in Section 6 when discussing the prob- Causality in Statistical Theo-
lem of necessity and chance in microphenomena that quantum mechanics ries
is a statistical theory. Hence, the question of causality in quantum mechan-
ics may be seen first of all from the point of view of the manifestation of
causality in general in statistical theories (as statistical mechanics, physi-
cal kinetics, microscopic electrodynamics).
When treating quantum mechanics as a statistical theory, one must Statistical Nature of Quantum
remember that it occupies a special place among these theories. Within Mechanics
the framework of classical physics the laws describing the behaviour of
large number of objects are of statistical nature, while the laws relating
to the behaviour of an individual object are dynamic. By considering the
element of chance in the behaviour of a single object, quantum mechanics
places itself in a special position – that of a statistical theory of an individ-
ual object. That is why we earlier called quantum mechanics a statistical
theory in principle.
(On the right-hand side the sum is taken over all the basic states.) The
expansion coefficients 𝑈𝑖 𝑗 (𝑡, Δ𝑡) for small values of Δ𝑡 will be represented
in the form
𝑖
𝑈𝑖 𝑗 (𝑡 + Δ𝑡) = 𝛿𝑖 𝑗 + 𝐻𝑖 𝑗 (𝑡) Δ𝑡 (12.6)
ℏ
This representation is justified by the fact that for Δ𝑡 → 0 the coefficient
𝑈𝑖 𝑗 will obviously be transformed into 𝛿𝑖 𝑗 (𝛿𝑖 𝑗 is the Kronecker delta
symbol).
𝐶𝑖 (𝑡 + Δ𝑡) − 𝐶𝑖 (𝑡) Õ
−𝑖ℏ = 𝐻 𝑖 𝑗 (𝑡) 𝐶 𝑗 (𝑡) (12.7)
Δ𝑡 𝑗
𝑑 Õ
−𝑖ℏ 𝐶𝑖 (𝑡) = 𝐻 𝑖 𝑗 (𝑡) 𝐶 𝑗 (𝑡) (12.8)
𝑑𝑡 𝑗
The matrix 𝐻 𝑖 𝑗 is called the Hamiltonian matrix. We make the follow- Hamiltonian Matrix
ing remarks concerning the Hamiltonian matrix:
(1) The time dependence of the Hamiltonian matrix reflects the depen-
dence of physical conditions on time (for example, a microparticle
situated in a magnetic field which varies with time). If the condi-
tions do not change, the matrix does not depend on time.
(2) If the Hamiltonian matrix is diagonalized (only its diagonal ele-
ments are non-zero) then in this case the elements of the matrix
120 b as ic c on c e pt s o f q ua n t u m m e ch a n ics
𝐻 𝑖 𝑗 = 𝐻 ∗𝑗 𝑖 (12.9)
𝑑 Õ
𝐶 𝑖 𝐶 ∗𝑖 = 0 (12.10)
𝑑𝑡
Taking into account that
𝑑 Õ Õ 𝑑𝐶 𝑖 Õ 𝑑𝐶 ∗
𝐶 𝑖 𝐶 ∗𝑖 = 𝐶 ∗𝑖 + 𝐶𝑖 𝑖
𝑑𝑡 𝑑𝑡 𝑖
𝑑𝑡
Finally, we note that from (12.9) one can deduce that the diagonal
elements of the Hamiltonian matrix are real numbers. This circumstance
is in conformity with the above-mentioned role of the diagonal elements
as values of the energy of a microparticle.
The answer to this question may be given by considering expression What Are the Requirements
(12.8) for the basic quantum-mechanical equation. The gist of the answer for a Causal Description of
is as follows. Firstly, we must choose a set of basic states { h 𝑖 | }; secondly, Phenomena in Quantum The-
we must find the form of the Hamiltonian matrix considered in the system ory?
of chosen basic states. After this we can make definite predictions by
using equation (12.8).
So, every time when we talk about a microparticle with two basic
states, it is assumed that we only consider the possible changes in some
particular “parameter” of the microparticle (for example, its polarization).
All the other “parameters” are assumed to be known. We shall now cite
some examples where one can speak of a microparticle with two basic
states. These states are denoted by h 1 | and h 2 |, respectively.
The ammonia molecule consists of one nitrogen atom and three hydro-
gen atoms. The nitrogen atom does not lie in the plane passing through
the hydrogen atoms (for brevity, we shall call this plane the 𝐻 -plane). The
state h 1 | corresponds to the nitrogen atom being on one side of the 𝐻 -
plane, and the state h 2 | corresponds to this atom being on the other side
of the 𝐻 -plane.
h 𝑠 | = 𝐶1 h 1 | + 𝐶2 h 2 | (13.1)
First case. The non-diagonal elements 𝐻 12 and 𝐻 21 are equal to zero (the
Hamiltonian matrix is diagonalized). In this case the system of
equations (13.2) splits into two independent equations
𝑑
−𝑖ℏ 𝐶 1 = 𝐻 11 𝐶 1
𝑑𝑡 (13.3)
𝑑
−𝑖ℏ 𝐶 2 = 𝐻 22 𝐶 2
𝑑𝑡
A Brief Interlude
aut hor: In Section 10, we were talking about the principle of su-
perposition of states. During the course of the discussion we did
not take into account the possibility of the development in time of
transition between the states giving rise to the superposition.
aut hor: First of all, let us expand (13.1) in the following form:
aut hor: That is true but don’t forget that when considering transi-
tions we should take into account not only the non-diagonality of
the Hamiltonian matrix, but also the superposition relation (13.1a).
In this case, everything depends on the nature of the time depen-
dence of the amplitudes 𝐶 1 and 𝐶 2 . In one case it so happens that
124 b as ic c on c e pt s o f q ua n t u m m e ch a n ics
We shall consider two cases exhibiting the time dependence of the Change of Amplitudes of State
amplitudes 𝐶 1 and 𝐶 2 . in Time
| 𝐶 1 (𝑡) | 2 = | 𝐶 1 (0) | 2
(13.5)
| 𝐶 2 (𝑡) | 2 = | 𝐶 2 (0) | 2
𝐻 12 = 𝐻 21 ≡ −𝐴 (13.7)
𝑑
−𝑖ℏ 𝐶 1 = 𝐸 0 𝐶 1 − 𝐴𝐶 2
𝑑𝑡 (13.8)
𝑑
−𝑖ℏ 𝐶 2 = 𝐸 0 𝐶 2 − 𝐴𝐶 1
𝑑𝑡
𝑑
−𝑖ℏ (𝐶 1 + 𝐶 2 ) = (𝐸 0 − 𝐴) (𝐶 1 + 𝐶 2 )
𝑑𝑡
𝑑
−𝑖ℏ (𝐶 1 − 𝐶 2 ) = (𝐸 0 + 𝐴) (𝐶 1 − 𝐶 2 )
𝑑𝑡
Let us compare the expressions for 𝐶 1 + 𝐶 2 and 𝐶 1 − 𝐶 2 , obtained in the Diagonalization of the Hamil-
previous section, with (13.4). This comparison allows us to conclude that tonian Matrix
the amplitudes 𝐶 1 + 𝐶 2 and 𝐶 1 − 𝐶 2 describe the stationary states of the
microparticle with energies equal to (𝐸 0 − 𝐴) and (𝐸 0 − 𝐴), respectively.
Further, we introduce a new pair of basic states
1
h I | = √ (h 1 | − h 2 |)
2
(13.12)
1
h II | = √ (h 1 | + h 2 |)
2
(it is easy to see that if the states h 1 | and h 2 | satisfy the orthonormaliza-
tion condition (10.8), the states h I | and h II | also satisfy this condition). By
using (13.12), we rewrite (13.1) in the form
𝐶1 − 𝐶2 𝐶1 + 𝐶2
h𝑠 | = √ hI| + √ h II | (13.13)
2 2
It can be seen from here that a transition from the basic states h 1 | and
h 2 | to the basic states h I | and h II | corresponds to a transition from am-
ph y si c al f o u n dat i o n s o f q ua n t u m m e c h a n i c s 127
𝐶1 − 𝐶2 𝐶1 + 𝐶2
plitudes 𝐶 1 and 𝐶 2 to amplitudes √ and √ . Since the
2 2
latter describe the stationary states of the microparticle, it follows that the
transition under consideration is associated with a diagonalization of the
Hamiltonian matrix.
" # " #
𝐸0 −𝐴 𝐸0 + 𝐴 0
→
−𝐴 𝐸0 0 𝐸0 − 𝐴
In the general case the non-diagonal elements of the Hamiltonian General Case
matrix are different from zero and so the simplifying conditions (13.6) and
(13.7) are not applicable. In this case one must solve not the simplified
system of equations (13.8), but the more general system of equations (13.2)
for a microparticle with two basic states. We suggest that the reader, if
he so desires, solves the system (13.2) himself, assuming for the sake of
simplicity that the Hamiltonian matrix is invariant in time12 We shall 12 Such a solution is given, for example,
in
limit ourselves here giving some results. Feynman, R. P., Leighton, R. B., and
Sands, M. (1965). Lectures on Physics,
The energy of the stationary states of a microparticle is determined by volume 3. Addison-Wesley Reading, Mass
the expression
s 2
𝐻 11 + 𝐻 22 𝐻 11 − 𝐻 22
𝐸 I, II = ± + 𝐻 12 𝐻 21 (13.14)
2 2
h I | = 𝑎1 h 1 | + 𝑏1 h 2 |
(13.15)
h II | = 𝑎 2 h 1 | + 𝑏 2 h 2 |
128 b as ic c on c e pt s o f q ua n t u m m e ch a n ics
where
| 𝑎1 |2 + | 𝑏1 |2 = | 𝑎2 |2 + | 𝑏2 |2 = 1
𝑎1 𝐻 12
= (13.16)
𝑏 1 𝐸 I − 𝐻 11
𝑎2 𝐻 21
=
𝑏 2 𝐸 II − 𝐻 22
It can be easily seen that if 𝐻 11 = 𝐻 22 = 𝐸 0 and 𝐻 12 = 𝐻 21 = −𝐴,
the result (13.14) gives 𝐸 I, II = 𝐸 0 ± 𝐴, the superpositions (13.15) turn
into (13.12). In other words, we arrive at the simplified case of the non-
diagonal Hamiltonian matrix discussed above in detail. But if 𝐻 12 = 𝐻 21 =
0, the result (13.14) gives 𝐸 I = 𝐻 11, 𝐸 II = 𝐻 22 - we arrive at the case of the
diagonal Hamiltonian matrix (h I | = h 1 | , h II | = h 2 |).
We recall that the basic states = h 1 | and = h 2 | of the ammonia Example of the Ammonia
molecule were chosen using graphic physical considerations: they cor- Molecule
respond respectively to the position of the nitrogen atom on one side of
the 𝐻 -plane and on the other. Since these positions are symmetrical, we
may take 𝐻 11 = 𝐻 22 ≡ 𝐸 0 . Assuming further that the elements 𝐻 12 and
𝐻 21 are real (𝐻 12 = 𝐻 21 ≡ −𝐴), which, as it turns out, does not involve
the loss of generality in this case, we arrive at the. situation to which the
simplified system of equation (13.8) corresponds. It follows from this that
the energy levels of a molecule are essentially 𝐸 0 + 𝐴 and 𝐸 0 − 𝐴. We
emphasize that if no transitions took place between the states h 1 | and
h 2 |, there would have been only one level 𝐸 0 in place of the levels 𝐸 0 + 𝐴
and 𝐸 0 − 𝐴. It would have been doubly degenerate since there would be
two states corresponding to it. It may be said that transitions between
the states h 1 | and h 2 | (associated with “pushing” of the nitrogen atom
through the 𝐻 -plane) correspond to a removal of degeneracy, i.e. to a
splitting of the level 𝐸 0 into two levels 𝐸 0 + 𝐴 and 𝐸 0 − 𝐴.
𝐻 11 = 𝐸 0 + E 𝑑
(13.17)
𝐻 22 = 𝐸 0 − E 𝑑
Now the positions of the nitrogen atom on either side of the 𝐻 -plane are
no longer physically symmetrical (𝐻 11 ≠ 𝐻 22 ). Assuming that 𝐻 12 = 𝐻 21 =
−𝐴 as before, we write the system of equations (13.2) for the case under
consideration:
𝑑
−𝑖ℏ 𝐶 1 = (𝐸 0 + E 𝑑) 𝐶 1 − 𝐴𝐶 2
𝑑𝑡 (13.18) Figure 13.2: Qualitative dependence of
𝑑
−𝑖ℏ 𝐶 2 = −𝐴 𝐶 1 + (𝐸 0 − E 𝑑) 𝐶 2
energy levels of ammonia molecule on the
𝑑𝑡 applied electric field intensity.
By using (13.14),we obtain the following expressions for the energy levels
ph y si c al f o u n dat i o n s o f q ua n t u m m e c h a n i c s 129
First case: the magnetic field is directed along the 𝑧-axis (𝐵𝑥 = 𝐵 𝑦 = 0).
In this case the basic states h 1 | and h 2 | are stationary; the state
h 1 | has energy −𝜇 𝐵𝑧 and the state h 2 | has energy 𝜇 𝐵𝑧 (𝜇 denotes
the magnetic moment of the electron). The amplitudes 𝐶 1 and 𝐶 2
satisfy two independent equations of the type (13.3):
𝑑
𝑖ℏ 𝐶 1 = 𝜇 𝐵𝑧 𝐶 1
𝑑𝑡 (14.1)
𝑑
−𝑖ℏ 𝐶 2 = 𝜇 𝐵𝑧 𝐶 2
𝑑𝑡
The Hamiltonian matrix of the electron is of the form
" #
h i −𝜇𝐵𝑧 0
𝐻 = (14.2)
0 𝜇𝐵𝑧
q
𝐵 = 𝐵𝑧 , in this case we must have 𝐵 = 𝐵𝑥2 + 𝐵 2𝑦 + 𝐵𝑧2 . Thus
q
𝐸 I = −𝜇 𝐵𝑥2 + 𝐵 2𝑦 + 𝐵𝑧2
q (14.3)
𝐸 II = 𝜇 𝐵𝑥2 + 𝐵 2𝑦 + 𝐵𝑧2
Note that
𝐸 I = −𝐸 II (14.4)
𝐻 11 = −𝜇 𝐵𝑧 ,
𝐻 22 = 𝜇 𝐵𝑧 ,
∗
𝐻 21 = 𝐻 12 = −𝜇 (𝐵𝑥 + 𝑖𝐵 𝑦 )
By using (14.6) we write the system of equation (13.2) for the case
under consideration:
𝑑
−𝑖ℏ 𝐶 1 = −𝜇 𝐵𝑧𝐶 1 + (𝐵𝑥 − 𝑖𝐵 𝑦 )𝐶 2
𝑑𝑡 (14.7)
𝑑
−𝑖ℏ 𝐶 2 = −𝜇 (𝐵𝑥 + 𝑖𝐵 𝑦 )𝐶 1 − 𝐵𝑧𝐶 2
𝑑𝑡
Let the direction of the magnetic field be determined by the polar angle Projection Amplitudes
𝜃 and azimuth 𝜑 (Figure 14.1). We shall assume that the electron spin is
directed along the field; consequently, the electron is in the stationary
state h I | with energy 𝐸 1 = −𝜇𝐵. According to (13.15), the state h I | can be
represented as a superposition:
h I | = 𝑎1 h 1 | + 𝑏1 h 2 | (14.8)
ph y si c al f o u n dat i o n s o f q ua n t u m m e c h a n i c s 131
𝑎1 𝐻 12
=
𝑏 1 𝐸 I − 𝐻 11 (14.9)
| 𝑎1 | + | 𝑏1 | = 1
2 2
𝐻 11 = −𝜇 𝐵𝑧 = 𝜇𝐵 cos 𝜃
(14.10)
𝐻 12 = −𝜇 (𝐵𝑥 − 𝑖𝐵 𝑦 ) = −𝜇𝐵 sin 𝜃 exp(−𝑖𝜑)
Figure 14.1: Direction of the magnetic
field.
Substituting (14.10) into (14.9), we find that
Note that (14.11) does not contain the magnetic induction 𝐵. It is ob-
vious that the result (14.11) must be valid in the limiting case 𝐵 → 0. In
other words we may exclude the field from consideration and interpret
(14.11) in the following way. It is clear that the direction of the electron
spin is determined by the angles 𝜃 and 𝜑. In this case the amplitude of the
probability that the electron spin is along the 𝑧-axis is 𝑎 1 and the ampli-
tude of the probability that the electron spin is in the opposite direction is
𝑏 1 . Expression (14.8) should be treated in this case as an expansion of the
spin state h𝜃, 𝜑 | in terms of the spin states h 𝑧 | and h −𝑧 |:
𝜃 𝑖𝜑 𝜃 𝑖𝜑
h𝜃, 𝜑 | = cos exp − h 𝑧 | + sin exp h−𝑧 | (14.12)
2 2 2 2
Let the direction of the electron spin be given by the angles 𝜃 and 𝜑 Precession of the Electron Spin
(the electron is in the state h𝜃, 𝜑 | ). This state can be represented in the
form of superposition (14.12) of the states h 𝑧 | and h−𝑧 |. Suppose that at
time 𝑡 = 0 we switch on a magnetic field 𝐵 which is directed along the
𝑧-axis. Now the states h 𝑧 | and h−𝑧 | become stationary states. Using this,
we write [see (13.4)]
−𝑖𝜇𝐵𝑧 𝑡
h𝜃, 𝜑 | 𝑧 i = 𝐶 1 exp
ℏ
(14.14)
𝑖𝜇𝐵𝑧 𝑡
h𝜃, 𝜑 | −𝑧 i = 𝐶 2 exp
ℏ
It follows from this that in time 𝑡 after the magnetic field has been
switched on the projection amplitudes assume the form
𝜃 𝑖 2𝜇𝐵𝑧 𝑡
h𝜃, 𝜑 (𝑡) | 𝑧 i = cos exp − 𝜑 +
2 2 ℏ
(14.16)
𝜃 𝑖 2𝜇𝐵𝑧 𝑡
h𝜃, 𝜑 (𝑡) | −𝑧 i = sin exp 𝜑+
2 2 ℏ
Thus, the switching on of a magnetic field along the 𝑧-axis does not
change the polar angle 𝜃 but changes the azimuth 𝜑, the change in 𝜑
being proportional to the time interval 𝑡 which elapses after switching
on the field. This means that the spin of an electron precesses around the
𝑧-axis (around the direction of magnetic field) with a constant angular
velocity. It can be easily seen that the angular velocity of the spin preces-
sion is given by the relation
2𝜇𝐵𝑧
𝜔= (14.17)
ℏ
We move one step further by ignoring coordinate axes. Suppose that
at time 𝑡 = 0 the direction of electron spin forms an angle 𝜃 with the
direction of the magnetic field. This angle will remain constant with
time, but the electron spin will precess around the field direction with an
ph y si c al f o u n dat i o n s o f q ua n t u m m e c h a n i c s 133
angular velocity
2𝜇𝐵
𝜔= (14.17a)
ℏ
We further suppose that the magnetic field is varying with time (in the
general case, both the direction and the magnitude of the vector 𝐵® are
varying). A change in the field leads to a corresponding change in the
electron spin precession: a change in the magnitude of the magnetic field
results in a change in the angular velocity of precession, while a change
in the direction of the field causes a change in the direction around which
the precession takes place.
two-state problems.
We shall consider some two-level system with the basic states h 1 | and
h 2 |. We assign some vector to each state of the microparticle. A choice
of the basic states h 1 | and h 2 | in this case is equivalent to a choice of the
𝑧-axis (as if these two states correspond to the two 𝑧-projections of the
134 b as ic c on c e pt s o f q ua n t u m m e ch a n ics
electron spin). Suppose that the microparticle is in the state h𝑠 (0) | at the
initial moment of time. We assign to this state a vector whose direction is
determined by the angles 𝜃 (0) and 𝜑 (0):
In order to find the angles 𝜃 (0) and 𝜑 (0) we must expand the state h𝑠 (0) |
in terms of the basic states h 1 | and h 2 | and use for the coefficients of ex-
pansion expression (14.13) for the projection amplitudes. This expansion
is of the form
𝜃 (0) 𝑖𝜑 (0) 𝜃 (0) 𝑖𝜑 (0)
h𝑠 (0) | = cos exp − h 1 | + sin exp h2|
2 2 2 2
(14.19)
Further, let us turn to the Hamiltonian matrix of the microparticle. First
of all we shift the zero point of the energy in such a way that it is located
precisely half-way between the two energy levels or, in other words, so
that the condition (14.4) is satisfied. In this case
𝐻 11 + 𝐻 22 = 0 (14.20)
𝐻 11 = −𝜇 𝐵𝑧
(14.21)
𝐻 22 = −𝜇 (𝐵𝑥 − 𝑖𝐵 𝑦 )
𝑑
−𝑖ℏ 𝐶 I = −𝐴𝐶 I + E 𝑑𝐶 II
𝑑𝑡 (14.24)
𝑑
−𝑖ℏ 𝐶 II = E 𝑑𝐶 I − 𝐴𝐶 II
𝑑𝑡
This system is convenient for drawing the analogy with the electron in
a magnetic field. Comparing (14.24) and (14.7), we find that the quantity
𝐴 corresponds to −𝜇 𝐵𝑧 and the quantity E 𝑑 to −𝜇 𝐵𝑥 · Consequently, we
have to consider the precession of the vector describing the state h𝑠 | of
the molecule in the “magnetic field” which is made up of two components:
a constant component along the 𝑧-axis, associated with the effect of the
“throwing” of a nitrogen atom through the 𝐻 -plane, and a component
along the 𝑧-axis, associated with the electric field. The latter component
may, obviously, vary with time.
Pauli Spin Matrices. In conclusion, we shall mention the Pauli spin matrices which are
widely used in the quantum mechanics of two-level systems. These matrices are of the form
" # " # " #
𝑥 0 1 𝑦 0 −𝑖 𝑧 1 0
𝜎 = 𝜎 = 𝜎 = (14.25)
1 0 𝑖 0 0 −1
By using these matrices, we can rewrite the expression for the elements of Hamiltonian matrix of all
electron in a magnetic fiald (14.6) in the following form:
𝑦
𝐻𝑖 𝑗 = −𝜇 𝜎𝑖𝑥𝑗 𝐵𝑥 + 𝜎𝑖 𝑗 𝐵 𝑦 + 𝜎𝑖𝑧𝑗 𝐵𝑧 (14.26)
𝐻𝑖 𝑗 = −𝜇 𝜎®𝑖 𝑗 𝐵® (14.27)
Pauli spin matrices are useful because any second-order matrix (in particular, the Hamiltonian matrix
of any microparticle with two basic states) may be represented as a superposition of these matrices.
The Pauli spin matrices introduced for an electron in a magnetic field have proved to be convenient for
considering a wide range of two-level problems. This is not surprising when we consider the possibility,
discussed above, of generalizing the problem of an electron in a magnetic field to arbitrary two-level
systems.
Let h 𝑥 | be the state of a microparticle corresponding to its localization The Wave Function as the
at a point in space with the coordinate 𝑥 (for simplicity we consider the Amplitude of State
one-dimensional case). Then h 𝑠 |𝑥 i may be considered as the probability
amplitude that a microparticle in the state h 𝑠 | has the coordinate 𝑥.
𝜓𝑠 (𝑥) = h 𝑠 |𝑥 i (15.2)
the 𝑥-representation).
Wave functions are frequently used in practice in the 𝑥-representation Generalization of the Concept
(coordinate representation). However apart from the 𝑥-representation, of The Wave Function
other representations are obviously also possible. In this connection, the
concept of the wave function must be generalized:
𝜓𝑎 (𝛽) = h 𝛼 | 𝛽 i (15.4)
By giving the wave function 𝜓𝑎 (𝛽) we give the exact values of the
quantities in the 𝛼-set and probable values of the quantities in the 𝛽-set.
Correspondingly, by giving the function 𝜑𝑎 (𝛾) we give the exact values of
the 𝛼-set and probable for the values of the 𝛾-set. It could be said that the
function 𝜓𝑎 (𝛽) describes the state h 𝛼 | in the 𝛽-representation, while the
function 𝜑𝑎 (𝛾) describes the same state, but in the 𝛾-representation. The
fact that different functions 𝜓𝑎 (𝛽) and 𝜑𝑎 (𝛾) are used for describing the
same state h 𝛼 | , indicates that there must be some connection between
them. This connection is expressed through the principle of superposition
of states. Assuming that 𝛾-values change discretely, we can write
Õ
𝜓𝑎 (𝛽) = 𝜑𝑎 (𝛾𝑖 ) 𝜒𝛾𝑖 (𝛽) (15.5)
𝑖
It can be easily seen that (15.5) is the expression for the superposition of
amplitudes of states:
Õ
h 𝛼 |𝛽 i = h 𝛼 |𝛾𝑖 i h 𝛾𝑖 |𝛽 i (15.5a)
𝑖
Ψ𝑎𝑖 (𝛼 𝑗 ) = 𝛿𝑖 𝑗 (15.7)
Ψ𝛼 0 (𝛼) = 𝛿 (𝛼 − 𝛼 0) (15.8)
𝛿 (𝛼 − 𝛼 0) = 0 at 𝛼 ≠ 𝛼 0
∫∞ (15.9)
𝛿 (𝛼 − 𝛼 0) 𝑑𝛼 = 1
−∞
∫∞
𝑓 (𝛼) 𝛿 (𝛼 − 𝛼 0) 𝑑𝛼 = 𝑓 (𝛼 0) (15.10)
−∞
From (15.12) and (15.7), we obtain from (15.11) the condition for the or-
thonormalization of the eigenfunctions 𝜓𝛼𝑖 (𝛽)
∫
𝜓𝛼𝑖 (𝛽) 𝜓𝛼∗ 𝑗 (𝛽) 𝑑𝛽 = 𝛿𝑖 𝑗 (15.13)
As an example, we consider the case of a freely moving microparticle. The Wave Function of a Freely
For simplicity, we assume that it has zero spin. The wave function in Moving Microparticle
15 15 We shall derive this result later (see
coordinate (three-dimensional) representation has the form
Section 20).
𝑟 ) = 𝑝®0 𝑟®
𝜓𝑝®0 (®
(15.15)
− 23 𝑖 𝑝®0𝑟®
= (2𝜋ℏ) exp
ℏ
where 𝑝®0 is the momentum of the microparticle and 𝑟® is its spatial coordi-
ph y si c al f o u n dat i o n s o f q ua n t u m m e c h a n i c s 139
The expression “crazy theory” as one which is “crazy enough to be “Crazy Ideas”
true” was once coined by Bohr. This expression reflects the stunning im-
pression produced on Bohr’s contemporaries by the astonishing physical
discoveries made at the beginning of the 20th century, discoveries which
could not be confined within the framework of classical concepts. It be-
came obvious that an explanation of these discoveries required radically
new ideas and a new approach.
. . .the new lesson which has been impressed upon physicists stresses the
caution with which all usual conventions must be applied as soon as we
are not concerned with everyday experience . . .In the study of atomic
phenomena we have repeatedly been taught that questions which were
believed to have received long ago their final answers had most unexpected
surprises in store for us.
The revision of the concepts and the rejection of many accepted no- The Essence of Quantum Me-
tions could well be considered as a “negative aspect” of quantum mechan- chanics
ics. Let us now consider its “positive aspect”.
First: quantum mechanics showed that the basic laws of nature are not
dynamic but are statistical, and that the probabilistic form of causal-
ity is the fundamental form while the classical determinism is just
its limiting (degenerate) case.
Second: quantum mechanics revealed that probability in nature should
not be dealt with as in classical statistical theories. It was found that
in certain cases it is not the probabilities of events that should be
summed, but rather the amplitudes of these probabilities. This
leads to the interference of probability amplitudes.
As Born pointed out in Physics in my generation,17 the statistical meth- 17 Born, M. (1956). Physics In My Gen-
eration: A Selection of Papers. Pergamon
ods found wider applicability with the development of physics. As re- Press, London
gards modern physics, it is completely based on statistical foundations.
In Born’s view, it is the quantum theory that established the closest links
between statistics and the basic aspects of physics. This should be con-
sidered as an important event in the history of human knowledge, with
consequences reaching far beyond the limits of science. It is sometimes
said that the fundamental difference between quantum mechanics and
classical mechanics is determined by, the statistical nature of the former
and dynamic nature of the latter. Upon careful consideration, this appar-
ently bland and irrefutable statement turns out to be incorrect. While
revealing the pre-eminence of statistical laws in physics, quantum me-
chanics shows at the same time that dynamic laws with their unique
predictions are, as a matter of fact, a special (degenerate) case of prob-
142 b as ic c on c e pt s o f q ua n t u m m e ch a n ics
ability laws. In this respect not only quantum mechanics, but classical
mechanics as well, must be, strictly speaking, formulated in the language
of probabilities18 The qualitative difference between quantum mechanics 18 This point of view is systematically
analysed by Myakishev (1973) where
and classical mechanics (or classical physics in general) depends on how it is stated, in particular, that Feynman’s
the relations among probabilities are considered. It has been mentioned in concepts of path integrals in fact converts
the principle of least action into the princi-
Mayakishev the main difference between quantum mechanics and classi- ple of maximum probability, i.e, it proves
cal mechanics does not lie in the statistical nature of the former. It lies in that the fundamental dynamic principle is
essentially statistical in nature.
the fact that it is not the probability but its amplitude, the wave function, Myakishev, G. Y. (1973). Dynamic and
that is of primary importance in quantum mechanics. This leads to the Statistical Laws in Physics (in Russian).
interference of probabilities, an effect which does not have an analogy in Nauka, Moscow
classical mechanics.
Developing the above ideas, let us single out the following points:
According to Feynman, one of the most outstanding achievements of The Specific Nature of
quantum mechanics lies in the fact that it allows so much to be extracted Quantum-Mechanical De-
from so little. scription of Phenomena
The reader has already found out how much can be achieved from the
phenomenon of interference of amplitudes (see Section 9), on the basis of
the principle of superposition of states (see Section 10), and from a consid-
eration of the simplest quantum-mechanical systems, i.e, microparticles
with two basic states (see Section 13, 14). The relative formal simplicity of
the description of microphenomena is connected with the specific nature
of this description. Remember that for a quantum-mechanical description
we must know firstly the basic states and, secondly, the Hamiltonian ma-
trix, which reflects the physics of the phenomena under consideration. A
simplification in the description can be achieved because of the following
two circumstances.
It has been noted earlier (see Section 10) that in classical physics
all states of a particle should be considered as mutually orthogo-
nal, or, in other words, as basic states. Because of this, the above-
mentioned simplifying situation is impossible here in principle.
Quantum mechanics forces us to take a fresh look at the well-known Probability in Quantum Me-
theorem of addition of probabilities for incompatible events. We have chanics
to consider not only the incompatibility but also the distinguishability
of the events. This is where the novelty of the approach lies. It is well
known that in the probability theory used in classical physics, as well as
144 b as ic c on c e pt s o f q ua n t u m m e ch a n ics
𝜑 (𝜃 ) = h 𝑓1 | 𝑠 1 i h 𝑓2 | 𝑠 2 i
𝜑 (𝜋 − 𝜃 ) = h 𝑓2 | 𝑠 1 i h 𝑓1 | 𝑠 2 i
| h 𝑠1 | 𝑠2 i | = 1 (16.1)
𝑤 = | 𝜑 (𝜃 ) + 𝜑 (𝜋 − 𝜃 ) | 2 (16.2)
𝑤 = | 𝜑 (𝜃 ) | 2 +| 𝜑 (𝜋−𝜃 ) | 2 +| h 𝑠 1 | 𝑠 2 i | 2 (𝜑 (𝜃 ) 𝜑 ∗ (𝜋 − 𝜃 ) + 𝜑 ∗ (𝜃 ) 𝜑 (𝜋 − 𝜃 ))
(16.4)
h 𝑠1 | 𝑠2 i = 0 (16.5)
𝑤 = | 𝜑 (𝜃 ) | 2 + | 𝜑 (𝜋 − 𝜃 ) | 2 (16.6)
Thus we find that the theorem of the addition of probabilities “holds” only
in the third of the above-mentioned cases, i.e, in the case of completely
ph y si c al f o u n dat i o n s o f q ua n t u m m e c h a n i c s 145
It can be easily seen that the result (16.4) based on the addition of
probabilities as well as the-addition of amplitudes is the most general
one. When the condition (16.5) is satisfied, it at once leads to the “purely”
classical case of addition of probabilities while condition (16.1) is fulfilled
“purely” in the case of addition of amplitudes of probabilities.
We draw the reader’s attention to the following fact. In order to ex- Quantum Mechanics and
plain the interference results in experiments with microparticles (for ex- Interference
ample, the interference pattern on the detector-screen in Experiment 1 of
Section 7), we can formally proceed in two different ways. One way cor-
responds to the “conservation” in quantum mechanics of the theorem of
the addition of probabilities for any incompatible events. This way, how-
ever, requires a comparison of the microparticle with some classical wave.
The other way corresponds to the addition of probability amplitudes. In
this case an explanation of interference results no longer requires the
introduction of any visual wave model.
By the way, the last fact may be taken as the starting point for a gen-
eralization of the very concept of “wave process”. Such a generalization
assumes a transition from visual classical waves with real amplitudes to
some sort of generalized waves with complex amplitudes. The classical
waves must appear as an extreme (degenerate) case of such generalized
waves. In other words, quantum-mechanical interference may be used
for an extension of the framework of the accepted wave picture (which,
incidentally, is invariably accompanied by rejection of a graphic represen-
tation) and for creating a theory for the generalized wave processes which
would reflect not only the probability nature of physical laws but also the
special relations among probabilities in nature.
A Brief Interlude
rea de r: It is not clear what you wanted to say in the last sentences
which, though quite eloquent, are tentative. Please explain them, if
possible.
rea de r: But can one perceive in this example any tendency towards
the development of modern physics?
aut hor: Let us take another example, that of a laser. We shall not
discuss the principle of its working here; it is just sufficient to men-
tion that it is based on some nonlinear effect, called the saturation
effect. Let us take another instrument, the second-harmonic gener-
ator (that is what a transformer of coherent light which doubles
the frequency is called in quantum mechanics). We shall simply
state that this instrument also is based on the principle of nonlin-
ear optical effect called the generation of second harmonics. Thus,
the laser produces coherent light of a definite frequency, while the
second-harmonic generator partially transforms the frequency of
this light. We can say that we first use the saturation effect and then
the effect of second-harmonic generation. Such is a general situa-
tion corresponding to a simple “summing” of these effects. Now,
suppose that both these effects are used simultaneously. In order
148 b as ic c on c e pt s o f q ua n t u m m e ch a n ics
The dialectical nature of quantum mechanics is reflected in its very Complementary Principle
initial principles. In this connection the principle of complementarity, put
forth by Bohr, is of special interest. This principle forms in fact the logical
foundation of the entire system of quantum-mechanical ideas.
equally essential aspects of all well-defined knowledge about the object. 21 21 N. Bohr, “On the Notions of Causality
and Complementarity”
. . .In atomic physics the word ‘complementarity’ is used to characterize
the relationship between experiences obtained by different experimental
arrangements and visualizable only by mutually exclusive ideas . . .22 22 N. Bohr, “Natural Philosophy and
Human Cultures”
Evidence obtained under different experimental conditions cannot be
comprehended within a single picture, but must be regarded as comple-
mentary . . .23 23 N. Bohr, “Discussion with Einstein
on Epistemological Problems in Atomic
Physics”
In quantum physics, however, evidence about atomic objects obtained
by different experimental arrangements exhibits a novel kind of comple-
mentary relationship. Indeed, it must be recognized that such evidence
which appears contradictory when combination into a single picture is
attempted, exhausts all conceivable knowledge about the object.24 24 N. Bohr, “Quantum Physics and
Philosophy”.
We advise the reader to carefully read the words of Bohr once again.
Thus, data about a microparticle may be “graphically interpreted” only on
the basis of “ideas mutually excluding one another” In this sense they can-
not be added in a simple way, and “cannot be contained in a single picture”.
Various data have “peculiar” (the reader must not let this epithet go unno-
ticed) relations with one another, hence the term “complementarity”. The
peculiarity of the “complementarity” relations lies in the fact that data
“complementary” to one another may be obtained only “under different
experimental conditions.”
It is true that dialectical nature is inherent in every physical science The Dialectical Nature of
to some extent. Still it may be stated that classical physics, because of Quantum Mechanics
the very style of its philosophy (unambiguous predictions in theories of
dynamic type, the approach to any object as a “combination” of certain
“details”, and to any phenomenon as a succession of certain elementary
events, etc.) is drawn towards metaphysics. In this sense the significance
of quantum mechanics cannot be overestimated. It has convincingly
shown that a higher level of knowledge of the laws of nature is inevitably
linked with a deeper and more serious knowledge and application of the
methods of materialistic dialectics.
stead constructs so abstract that the ordinary mind can never attain have been used in the discussion: (Bohr,
1958b, 1955; Dobrolyubov, 1948; Cooper,
25
them. 1968; Perrault, 1946).
Bohr, N. (1958b). Natural philosophy
aut hor: That is why I want to talk about quantum mechanics and human cultures. In Atomic Physics
and Human Knowledge, pages 23–31. New
and “common sense”. I have many times heard the complaint that
York, Wiley; Bohr, N. (1955). Science
quantum mechanics is hard to follow because its concepts are in and the unity of knowledge. In Leary,
“contradiction to common sense”. Unfortunately nobody knows L., editor, The Unity of Knowledge, pages
47–62. Doubleday; Dobrolyubov, N. A.
precisely the meaning of “common sense”. If I am not mistaken, you (1948). What is oblomovshchina? Literary
are of the view that “common sense” is a relative concept, and that Criticism, 2; Cooper, L. N. (1968). An
Introduction to the Meaning and Structure
its meaning changes significantly with the development in science.
of Physics. Harper and Row, N.Y; and
Perrault, C. (1946). The Sunshine Book
coo pe r: Yes, it is a cliché that the commonsense of the new genera-
tion is formed from concepts laboriously constructed by their elders,
that what is avant-garde for one generation is common sense and
prosaic for the next. It seems dubious that the Newtonian concep-
tion of the world would have been common sense to the Greeks in
the time of Aristotle, for that matter to the scholastic scholars. It
was not even so for many of his contemporaries. And those so en-
amored of common sense (at present Newton’s world) are often just
those who complain that the mechanical Newtonian view destroyed
the magical medieval world.
boh r: We all know the old saying that, if we try to analyse our own
emotions, we hardly possess them any longer, and in that sense
we recognise between physical experiences, for the descriptions of
which words such as “thoughts” and “feelings” are adequately used,
a complementary relationship similar to that between the experi-
ences regarding the behaviour of atoms obtained under different
experimental arrangements.
aut hor: But can’t we go one step further, and try to draw some
analogies between modern ideas in physics and ideas contained in
famous literary works? It would be quite interesting to compare
poetical truth and scientific truth.
aut hor: I remember that you drew analogy between Oblomov and
such literary heroes as Onegin, Pechorin, Rudin.
do b ro l uy b ov: Yes, indeed. But the point is that they all have one
common feature-a fruitless striving for activity, the consciousness
that they could do a great deal but will do nothing.
aut hor: (to Bohr) And what would you say to this?
Let us now consider the second literary example, the famous fairy-
tale “Cinderella” by Charles Perrault. Will you please narrate the
scene in which the fairy godmother sees Cinderella off to the ball at
King’s palace.
156 b as ic c on c e pt s o f q ua n t u m m e ch a n ics
per r ault: Here you are: The fairy godmother then said to Cin-
derella, “Here is a coach fit for the ball, and coachman and footmen.”
As Cinderella stepped into the carriage, her fairy godmother said,
“Remember, you must not stay a minute after twelve, for if you do,
your coach will become a pumpkin again, the horses will turn back
into mice; the footmen will become lizards, and the coachman will
become a rat; and your dress will turn to rags.”
aut hor: Thank you. I wanted to draw your attention to the fact that
the omnipotent fairy godmother gave the coach and the dress to
Cinderella only for a while, until midnight. And why not for good?
It is clear that she could have done that but it would be against the
inherent logic and the central idea of the tale. It would, so to say,
take away the “charm”. If it is for a while, well and good; but if it
is for ever, there is no charm. Doesn’t it remind us of a model of
virtual transitions? The conservation laws are violated, treasures are
created out of “nothing” with the help of a magic wand, but all this
is allowed only for a finite interval of time – “until midnight.” After
this, Cinderella must return to her previous state, and without the
beautiful dress. Just compare: a quantum system visits a new level
without any expenditure of energy from outside, only for a limited
interval of time after which the system must return to its previous
level. In the same way, Cinderella performs “virtual transitions”
between her house and the royal palace, enjoys and dances but is
careful not to overstep the agreed time limit.
Perrault: The messenger, who had been sent with the slipper,
said that everyone was to try it. He looked at Cinderella and saw
that she was beautiful. He ordered her to sit down and put the
slipper on It fitted her perfectly . . . Just then the fairy godmother ap-
peared and touched Cinderella with her wand and her rags became
a dress more beautiful than any she had yet worn.
aut hor: And so, it came true. The fairy godmother gave Cinderella
the dress to keep for ever. The virtual transition led to a real tran-
sition by Cinderella to “a new level”. The prince, the slipper, the
messenger - all played the role of the photon which, by interacting
with the system undergoing virtual transitions, has led to a real tran-
sition. Of course, the Cinderella story should not be seriously taken
as an illustration of the idea of virtual transitions, as an explanation
of quantum jumps. In the same way the novel Oblomov should not
be seriously taken as an illustration of the principle of superposition
of states or as an explanation of the problem of destruction of super-
ph y si c al f o u n dat i o n s o f q ua n t u m m e c h a n i c s 157
Every physical theory is a synthesis of certain physical ideas (advanced Some General Remarks
on the basis of experiment) and a certain mathematical apparatus. The
building-up of a theory is a complicated and controversial process devel-
oping according to successive approximations. However, in this contro-
versial process, at least in its initial stage, a completely definite logical
structure is envisaged, including three logically successive stages:
In Quantum Physics and Philosophy 1 , Bohr writes 1 Bohr, N. (1963). Essays 1958-1962 on
atomic physics and human knowledge. In-
terscience, N.Y., Interscience, NY
. . .in quantal formalism, the quantities by which the state of a physical sys-
tem is ordinarily defined are replaced by symbolic operators subjected to a
non-commutative algorism involving Planck’s constant. This procedure pre-
vents a fixation of such quantities to the extent which would be required
tor the deterministic description of classical physics, but allows us to de-
termine their spectral distribution as revealed by evidence about atomic
processes. In conformity with the non-pictorial character of the formal-
ism, its physical interpretation finds expression in laws, of an essentially
statistical type . . .
An operator, when applied to some function, transforms it into a new Linear Operators (Basic Defini-
function. The notation tions)
𝐿ˆ 𝜓 (𝑥) = 𝜑 (𝑥) (17.1)
means that the operator denoted the symbol 𝐿ˆ acts on the function 𝜓 (𝑥),
as a result of which we get function 𝜑 (𝑥).
𝐿ˆ (𝜓 1 + 𝜓 2 ) = 𝐿ˆ 𝜓 1 + 𝐿ˆ 𝜓 2
(17.2)
𝐿ˆ (𝑎𝜓 ) = 𝑎 𝐿𝜓
ˆ
The quantity 𝐿 (𝑥, 𝑦) is called the kernel of the operator. If the variable is
discrete, we will have instead of (17.3)
Õ
𝐿ˆ 𝜓𝑛 = 𝐿𝑛𝑚 𝜓𝑚 (17.4)
𝑚
The operator ê
𝐿 is said to be the transpose of the operator 𝐿ˆ if the following
condition is satisfied:
∫ ∫
Ψ(𝑥) 𝐿ˆ 𝜓 (𝑥) 𝑑𝑥 = 𝜓 (𝑥) ê
𝐿 Ψ(𝑥) 𝑑𝑥 (17.6)
li n e ar o p e r ato r s i n q ua n t u m m e c h a n i c s 163
e
𝐿 (𝑥, 𝑦) = 𝐿 (𝑦, 𝑥) (17.7)
e
𝐿 𝑛𝑚 = 𝐿 𝑛𝑚 (17.8)
Let us consider some linear operator 𝐿. ˆ Let us find its complex conjugate
∗
operator 𝐿ˆ ∗ . We then find the transposed operator ê
𝐿 for the operator 𝐿ˆ ∗ .
This operator is denoted by 𝐿ˆ † 2 and is said to be conjugate to the operator 2 In the original typesetting symbol + is
used to denote the conjugate, here we will
𝐿.
ˆ By using the concept of the conjugate operator, two important types of
be using † symbol.
linear operators are defined: Hermitian operators and unitary operators.
If
𝐿ˆ = 𝐿ˆ † (17.9)
𝐿ˆ 𝐿ˆ † = 𝐿ˆ † 𝐿ˆ = 1 (17.10)
The basic equation of the theory of linear operators has the form
𝐿ˆ 𝜓 = 𝜆 𝜓 (17.13)
The numbers 𝜆 for which the equation (17.13) has finite solutions, form
the spectrum of eigenvalues of the operator 𝐿.
ˆ The spectrum of eigenval-
ues of an operator may be continuous, discrete or mixed. The solutions
𝜓 (𝑥) of equation (17.13) are called the eigenfunctions of the operator 𝐿.
ˆ
One or more eigenfunctions may correspond to a given eigenvalue. If 𝑠
linearly independent eigenfunctions correspond to a certain value 𝜆1 it is
said that the eigenvalue 𝜆1 is 𝑠-fold degenerate.
We shall state three theorems concerning the basic Properties of Her- Properties of Hermitian Opera-
mitian Operators properties of Hermitian operators. (The proofs of the tors
first two theorems are given in Appendix A.)
164 b as ic c on c e pt s o f q ua n t u m m e ch a n ics
First Theorem
Second Theorem
Third Theorem
∗
In order to find 𝑐𝑛 , we multiply this equation by 𝜓𝑚 (𝑥) and inte-
grate with respect to 𝑥:
∫ Õ ∫
∗ ∗
𝜓𝑚 (𝑥) Φ (𝑥) 𝑑𝑥 = 𝑐𝑛 𝜓𝑚 (𝑥) 𝜓𝑛 (𝑥) 𝑑𝑥
𝑛
li n e ar o p e r ato r s i n q ua n t u m m e c h a n i c s 165
Thus, Õ
Φ (𝑥) = 𝑐𝑛 𝜓𝑛 (𝑥)
𝑛
where (17.18)
∫
𝑐𝑛 = 𝜓𝑛 (𝑥) Φ(𝑥) 𝑑𝑥
∗
In the case of a continuous spectrum, we must use the condition
(17.17). As a result, we get instead of (17.18)
∫
Φ (𝑥) = 𝑐 (𝜆) 𝜓𝜆 (𝑥) 𝑑𝜆
where (17.19)
∫
𝑐 (𝜆) = 𝜓𝜆 (𝑥) Φ(𝑥) 𝑑𝑥
∗
We shall mention one of the results which is a direct consequence
of the completeness of the system of eigenfunctions of a Hermitian
operator. Suppose
∫
Φ(𝑥) = 𝑐 (𝜆) 𝜓𝜆 (𝑥) 𝑑𝜆
and ∫
Ψ(𝑥) = 𝑏 (𝜆 0) 𝜓𝜆0 (𝑥) 𝑑𝜆 0
Suppose that the Hermitian operator 𝑀 transforms the function Φ(𝑥) Representations
into the function Ψ(𝑥) :
Ψ(𝑥) = 𝑀ˆ Φ(𝑥) (17.21)
It is said that (17.21) and (17.23) are two different representations of the
same transformation. The nature of the representation is determined by
the variables on which the initial and final functions depend. Hence we
speak of 𝑥-representation in the case of (17.21) and of 𝜆-representation
(representation of the operator 𝐿)
ˆ in the case of (17.23). Correspondingly,
the operator 𝑀ˆ in (17.21) is the operator of the given transformation,
defined in the 𝑥-representation [for convenience, we shall henceforth
write it as 𝑀ˆ (𝑥)], while the operator 𝑀ˆ (𝜆) occurring in (17.23) is the
operator of the given transformation, defined in 𝜆-representation.
Let us see what the Hermitian operator looks like in eigen representa-
tion. Let 𝜑 𝜇 (𝑥) be the eigenfunctions of the operator 𝑀.
ˆ In this case, we
can write (17.22) in the following form:
∫
Φ(𝑥) = 𝑐 (𝜇) 𝜑 𝜇 (𝑥) 𝑑𝜇
∫ (17.24)
Ψ(𝑥) = 𝑏 (𝜇) 𝜑 𝜇 (𝑥) 𝑑𝜇
whence we get
𝑀ˆ (𝜇) = 𝜇 (17.26)
Thus in its eigen representation, the Hermitian operator coincides with its
eigenvalues.
Multiplying both sides of the last of these equalities by 𝜓𝜆∗ (𝑥) and integrat-
ing with respect to 𝑥, we get
∫ ∫ ∫ ∫
0 ∗ 0 0
𝑐 (𝜆 ) 𝜓𝜆 (𝑥) 𝑀 (𝑥) 𝜓𝜆0 (𝑥) 𝑑𝑥 𝑑𝜆 =
ˆ 𝑏 (𝜆 ) 𝜓𝜆 (𝑥) 𝜓𝜆0 (𝑥) 𝑑𝑥 𝑑𝜆 0
∗
li n e ar o p e r ato r s i n q ua n t u m m e c h a n i c s 167
Lastly, by taking (17.23) into account, we finally get the result [compare
with (17.3)]: ∫
𝑀ˆ (𝜆) 𝑐 (𝜆) = 𝑀 (𝜆, 𝜆 0) 𝑐 (𝜆 0) 𝑑𝜆 0
where ∫
𝑀 (𝜆, 𝜆 0) = 𝜓𝜆∗ (𝑥) 𝑀ˆ (𝑥) 𝜓𝜆0 (𝑥) 𝑑𝑥 (17.27)
A transition from the 𝜆-representation to the 𝑥-representation is deter- Transition from One Represen-
mined with the help of (17.22). In operator form, these relations can be tation to Another as Unitary
written as Transformation
Φ(𝑥) = 𝑈 (𝑥, 𝜆) 𝑐 (𝜆)
(17.29)
Ψ(𝑥) = 𝑈 (𝑥, 𝜆) 𝑏 (𝜆)
Next, we put
∫ ∫
∗
Ψ (𝑥) Φ(𝑥) 𝑑𝑥 = Ψ∗ (𝑥) 𝑈ˆ (𝑥, 𝜆) 𝑐 (𝜆) 𝑑𝑥
∫
∗
= 𝑐 (𝜆) 𝑈ˆ † (𝜆, 𝑥) Ψ(𝑥) 𝑑𝜆
and, consequently,
𝑈ˆ (𝑥, 𝜆) 𝑈ˆ † (𝜆, 𝑥) = 1 (17.30)
Thus, the relations (17.29) describe the transition from one representa-
tion of functions to another carried out with the help of a unitary opera-
tor. Relation (17.32) describes the same transition for operators.
Quantities and properties which do not change with unitary transfor- Unitary Invariants
mations and are, consequently, independent of the choice of representa-
tion are called unitary inuariants. Unitary invariants include:
Note that unitary invariance of these integrals means that the follow-
ing relations hold:
∫ ∫
Ψ∗ 𝑀ˆ (𝑥) Φ(𝑥) 𝑑𝑥 = 𝑏 ∗ (𝜆) 𝑀ˆ (𝜆) 𝑐 (𝜆) 𝑑𝜆 (17.33)
∫ ∫
∗ ˆ𝑛
Ψ 𝑀 (𝑥) Φ(𝑥) 𝑑𝑥 = 𝑏 ∗ (𝜆) 𝑀ˆ 𝑛 (𝜆) 𝑐 (𝜆) 𝑑𝜆 (17.34)
li n e ar o p e r ato r s i n q ua n t u m m e c h a n i c s 169
Two operators 𝐿ˆ and 𝑀ˆ are called commutative if for any bounded Commutation of Operators
function Φ(𝑥) satisfy the following condition: and the System of Common
Eigenfunctions
𝑀ˆ 𝐿ˆ Φ(𝑥) = 𝐿ˆ 𝑀ˆ Φ(𝑥) (17.35)
If there is even one function for which (17.35) does not hold, the operators
𝐿ˆ and 𝑀ˆ are called non-commutative. The notation
[𝑀,
ˆ 𝐿]
ˆ = 𝑀ˆ 𝐿ˆ − 𝐿ˆ 𝑀ˆ
[𝑀,
ˆ 𝐿]
ˆ =0
𝑀ˆ 𝐿ˆ 𝜓𝜆𝜇 = 𝜆 𝑀ˆ 𝜓𝜆𝜇
= 𝜆 𝜇 𝜓𝜆𝜇
𝐿ˆ 𝑀ˆ 𝜓𝜆𝜇 = 𝜇 𝐿ˆ 𝜓𝜆𝜇
= 𝜇 𝜆 𝜓𝜆𝜇
(𝑀ˆ 𝐿ˆ − 𝐿ˆ 𝑀)
ˆ 𝜓𝜆𝜇 = 0
and since the functions 𝜓𝜆𝜇 (𝑥) are known to form a closed system, for any
function Φ(𝑥) we can write
Õ
( 𝑀ˆ 𝐿ˆ − 𝐿ˆ 𝑀)
ˆ Φ(𝑥) = 𝑐 𝜆𝜇 ( 𝑀ˆ 𝐿ˆ − 𝐿ˆ 𝑀)
ˆ 𝜓𝜆𝜇 (𝑥)
=0 Q.E.D.
170 b as ic c on c e pt s o f q ua n t u m m e ch a n ics
We turn to equation (12.8) discussed in Section 12. There we chose a The Influence of the Choice of
certain system of basic states of microparticle {h 𝑖 |}. An arbitrary state Basic States
h 𝑠 (𝑡)| of this microparticle at time 𝑡 was represented as a superposition of
these basic states:
Õ
h 𝑠 (𝑡)| = h 𝑠 (𝑡) | 𝑖i h 𝑖 | (18.1)
𝑖
The notation 𝐶𝑖 (𝑡) was employed for the amplitudes h 𝑠 (𝑡) | 𝑖 i. It was
shown that the amplitudes 𝐶𝑖 (𝑡) satisfy the equation
𝑑 Õ
−𝑖ℏ 𝐶𝑖 (𝑡) = 𝐻𝑖 𝑗 (𝑡) 𝐶 𝑗 (𝑡) (18.2)
𝑑𝑡
which permits one to find, from a knowledge of the amplitudes 𝐶𝑖 (𝑡)
and the Hamiltonian matrix 𝐻𝑖 𝑗 (𝑡) at time 𝑡, the amplitudes 𝐶𝑖 at any
subsequent moments of time.
The expression (18.1) clearly shows that the set of amplitudes {𝐶𝑖 (𝑡)}
depends on the choice of the system of basic states {h 𝑖 |}. Suppose we
have to go over to a new system of basic states {h 𝑚 |}. In order to accom-
plish this transition, we represent the earlier basic states in the form of
superposition of new basic states:
Õ
h𝑖 | = h𝑖 | 𝑚 i h𝑚 | (18.3)
𝑚
li n e ar o p e r ato r s i n q ua n t u m m e c h a n i c s 171
or
h 𝑠 (𝑡)| = 𝑐𝑚 (𝑡) h 𝑚 | (18.4)
where
Õ
𝑐𝑚 (𝑡) = 𝐶𝑖 (𝑡) h 𝑖 | 𝑚 i (18.5)
𝑖
We shall show how the old Hamiltonian matrix can he expressed in terms
of the new one. Substituting (18.5) in (18.6), we find
𝑑 Õ ÕÕ
−𝑖ℏ 𝐶 𝑗 0 (𝑡) h 𝑗 0 | 𝑚 i = 𝐻𝑚𝑛 (𝑡) 𝐶 𝑗 (𝑡) h 𝑗 | 𝑛 i
𝑑𝑡 𝑗 0 𝑛 𝑗
we can simplify the left-hand side of the last equation. As a result, we get
𝑑 Õ
−𝑖ℏ 𝐶𝑖 (𝑡) = 𝐻𝑖 𝑗 (𝑡) 𝐶 𝑗 (𝑡)
𝑑𝑡 𝑗
where
ÕÕ
𝐻𝑖 𝑗 (𝑡) = 𝐻𝑚𝑛 (𝑡) h 𝑗 | 𝑛 i h 𝑚 | 𝑖 i (18.7)
𝑛 𝑚
number of cases. Thus, in Section 13, the transition from the basic states
h 1 | and h 2 | to the basic states h I | and h II | was motivated by a desire to
diagonalize the Hamiltonian matrix, while in Section 14 it was done in
order to express the Hamiltonian matrix in a form convenient for drawing
an analogy with an electron in magnetic field.
Let us consider operator 𝐻ˆ (𝑡) satisfying the following relation: Transformation of an Equation
Expressing Causality Into
𝐻𝑖 𝑗 (𝑡) = h 𝑗 | 𝐻ˆ (𝑡) | 𝑖 i (18.8)
a Form Independent of the
The operator 𝐻ˆ (𝑡), acting on the basic state h 𝑖 |, gives rise to a new state Choice of Basic States
h𝜓 (𝑡) | = 𝐻ˆ (𝑡) | 𝑖 i, which is not basic. The element of the Hamiltonian
matrix 𝐻𝑖 𝑗 (𝑡) here plays the role of the amplitude h 𝑗 | 𝜓 (𝑡) i, i.e. of the
amplitude probability that a microparticle in the basic state h 𝑗 | may be
found in the state h𝜓 (𝑡)|. Substituting (18.8) in (18.2), we get
𝑑 Õ
−𝑖ℏ h 𝑠 (𝑡) | 𝑖 i = h 𝑠 (𝑡) | 𝑗 i h 𝑗 | 𝐻ˆ (𝑡) | 𝑖 i (18.9)
𝑑𝑡 𝑗
or
𝑑
−𝑖ℏ h 𝑠 (𝑡) | 𝑖 i = h 𝑠 (𝑡) | 𝐻ˆ (𝑡) | 𝑖 i (18.10)
𝑑𝑡
Taking the complex-conjugate of the equation and considering (9.33) and
(12.9), we get
𝑑
−𝑖ℏ h 𝑖 | 𝑠 (𝑡) i = h 𝑖 | 𝐻ˆ (𝑡) | 𝑠 (𝑡)i (18.11)
𝑑𝑡
or
𝑑
−𝑖ℏ | 𝑠 (𝑡) i = 𝐻ˆ (𝑡) | 𝑠 (𝑡) i (18.12)
𝑑𝑡
Note that while going over from (18.9) to (18.10) or from (18.11) to (18.12),
we have made use of the rules given in Section 10 under the heading
“The Mechanics of Quantum Mechanics”. The final equation (18.12) is
analogous to equation (18.2), but unlike the latter, it is independent of the
choice of basic states.
Each cell of the table contains two expressions having the same mean-
ing (indicated in brackets). However, the upper expression depends on
the choice of the system of basic states or coordinate axes, while the lower
one is independent of any such choice.
It is worthwhile giving an example for the lower right cell of the above
table, which shows how one can imagine an operator acting on a vector.
As an example of this (to which, of course, no “quantum-mechanical
meaning” should be assigned) let us consider the case when the matrix 𝐻𝑖 𝑗
is of the form
𝜕 𝜕
© 0 − ª
𝜕𝑟 3 𝜕𝑟 2 ®
®
𝜕 𝜕 ®
𝐻 = 0 − ®
𝜕𝑟 3 𝜕𝑟 1 ®
𝜕 𝜕 ®
− 0 ®
« 𝜕𝑟 2 𝜕𝑟 1 ¬
where 𝑟 1, 𝑟 2, 𝑟 3 are three Cartesian coordinates in space. In this case the
relation 𝑐® = 𝐻ˆ 𝑎® acquires the form which is familiar to those who have
® × 𝑎.
studied vector analysis: 𝑐® = ∇ ®
174 b as ic c on c e pt s o f q ua n t u m m e ch a n ics
Let us demonstrate some of the advantages of the operator approach. We Average Energy
shall show how the mean value h 𝐸 i of the energy of a microparticle in
some state h 𝑠 | is determined. Let {h 𝑖 |} be the basic states with energies
𝐸𝑖 . This means (see Section 13) that the Hamiltonian matrix is diagonal.
Hence,
h 𝑗 | 𝐻ˆ | 𝑖 i = 𝛿𝑖 𝑗 𝐸𝑖
or
h 𝑗 | 𝐻ˆ | 𝑖 i = h 𝑗 | 𝑖 i 𝐸𝑖 (18.13)
and try to find the mean value of the energy h 𝐸 i with the help of a for-
mula of the type (12.3):
Õ
h𝐸 i = | h 𝑠 | 𝑖 i | 2 𝐸𝑖 (18.14)
𝑖
where
Õ
|𝜑 i = | 𝑖 i 𝐸𝑖 h 𝑖 | 𝑠 i (18.16)
𝑖
𝐻ˆ | 𝑖 i = | 𝑖 i 𝐸𝑖 (18.17)
h 𝐸 i = h 𝑠 | 𝐻ˆ | 𝑠 i (18.18)
It can be seen from (18.18) that mean value of the energy of a microparti-
cle in the state h 𝑠 | is expressed only through the operator 𝐻ˆ . Basic states
do not enter this relation.
It has been mentioned above that the Hamiltonian matrix could be called Energy Operator (Hamilto-
the energy matrix (remember that the elements of the diagonalized Hamil- nian)
tonian matrix are essentially the possible values of the energy of the
microparticle). The connection between the operator 𝐻ˆ and the Hamilto-
nian matrix as well as relation (18.18), expressing the average energy of a
microparticle in terms of the operator 𝐻ˆ , justify the name energy operator
given to it. In the literature the operator it is also called the Hamiltonian.
𝑑
𝑖ℏ | 𝑠 (𝑡) i = 𝐻ˆ | 𝑠 (𝑡) i (18.12)
𝑑𝑡
h 𝐸 i = h 𝑠 | 𝐻ˆ | 𝑠i (18.18)
𝐻ˆ | 𝑓 i = | 𝑓 i 𝐸 (18.20)
Finally, we note that the Hamiltonian (as well as any other operator)
may act not only on the state | 𝑠 i, but also on its amplitude h 𝑖 | 𝑠 i, since
we always have the representation [see (17.4)]
Õ
𝐻ˆ 𝐶𝑖 (𝑡) = 𝐻𝑖 𝑗 𝐶 𝑗 (𝑡) (18.21)
𝑗
Using (18.21) and taking into account that h 𝑖 | 𝑠 i = 𝐶𝑖∗ , we can rewrite
(18.2) in a form which, as can be easily seen, is completely analogous in
form to (18.12):
𝑑
𝑖ℏ h 𝑖 | 𝑠 (𝑡) i = 𝐻ˆ h 𝑖 | 𝑠 (𝑡) i (18.22)
𝑑𝑡
176 b as ic c on c e pt s o f q ua n t u m m e ch a n ics
𝐻ˆ h 𝑖 | 𝑓 i = 𝐸 h 𝑖 | 𝑓 i (18.23)
The following two points must be noted when considering the role of Role of Operators in Quantum
linear operators in quantum mechanics. Mechanics
Let us consider the first point in detail. It means that besides the energy
operator 𝐻ˆ , other “physical operators” like the coordinate operator 𝑟®ˆ the
momentum operator 𝑝, ®ˆ the angular momentum operator 𝑀, ®ˆ etc. must be
introduced. In this respect, it is significant that the well-known dynamic
relations of classical mechanics may be transferred to quantum mechanics
in the same form, if we replace the physical quantities in these relations
by the corresponding Hermitian operators. In other words, the apparatus
of quantum mechanics may be built up in analogy with the apparatus
of classical mechanics, if we replace the dynamic variables with their
corresponding Hermitian operators. As an example, let us compare the
following expressions:
𝑝2 𝑝ˆ 2
𝐸= +𝑈 𝐻ˆ = + 𝑈ˆ (19.1)
2𝑚 2𝑚
ˆ
𝑀® = 𝑟® × 𝑝® 𝑀® = 𝑟®ˆ × 𝑝®ˆ (19.2)
2
Thus if 𝐴ˆ and 𝐵ˆ do not commute, then 𝐴ˆ + 𝐵ˆ ≠ 𝐴ˆ2 + 2𝐴ˆ𝐵ˆ + 𝐵ˆ 2 . In this
2
case 𝐴ˆ + 𝐵ˆ = 𝐴ˆ2 + 𝐴ˆ𝐵ˆ + 𝐵ˆ𝐴ˆ + 𝐵ˆ 2 . Besides, there are operators in quantum
mechanics which do not have classical analogies (for example, the spin
operator).
Let us consider the question: In exactly what way can we compare a Basic Postulates
physical quantity with a Hermitian operator? In other words, what is the
meaning of the word “compare” here? The following two basic postulates
provide an answer to this question.
When applied to the energy operator, (19.3) takes the form of (18.23).
A study of equation (18.23) permits us to find the possible values of the
energy of a microparticle and the corresponding amplitude values of the
stationary states.
The postulates formulated above “knit together” the physical and math- Mathematical Results and
ematical aspects; they “load” the mathematical symbols and inferences Their Physical Meaning
with a definite physical meaning. We shall demonstrate this with a num-
178 b as ic c on c e pt s o f q ua n t u m m e ch a n ics
ber of observations.
The last remark will be rigorously proved below. Here, we shall just
mention some ideas of a qualitative nature for this purpose. If operators
𝐻ˆ and 𝐿ˆ commute, the quantities 𝐸 and 𝑙 can be simultaneously measured
since there are states in which both these quantities have definite values.
A state in which energy has a definite value is stationary, i.e. has an
infinitely long “life” time. But this means that the quantity 𝑙 must also be
conserved for an infinitely long time, just like any other physical quantity
for which the given state is an eigenfunction.
If the quantity 𝑙 is measured in a state described by the amplitude Mean Value of a Quantity
h 𝜆 | 𝛼 i, then, according to basic postulates, the measured value will be 𝜆.
We assume now that the quantity 𝑙 is measured not in its eigenstate, but
in some “other” state, for example, in the state described by the amplitude
Φ𝑠 (𝛼) = h 𝑠 | 𝛼 i. In this case the result of a single measurement cannot be
predicted unambiguously; probabilistic predictions enter into the picture
now, thus permitting an estimation of the mean value h 𝜆i from a rela-
tively large number of measurements (in this connection see Section 12).
We shall show how to compute the mean value h 𝜆 i in the state h 𝑠 |, if
we know the Hermitian operator 𝐿ˆ corresponding to the quantity 𝑙, Note
that in the particular case when energy is used in place of the quantity 𝑙,
this problem was considered in Section 18, where the following result was
obtained:
h 𝐸 i = h 𝑠 | 𝐻ˆ | 𝑠 i (18.18)
(we assume for definiteness in this case that the variable 𝛼, which charac-
terizes the representation varies continuously). Let us prove this.
The result (19.4) is very important. In fact this one result is sufficient
to demonstrate the usefulness of the application of operators in quantum
mechanics.
In analogy with (18.18), the result (19.4) can be written in a more ab-
stract form which avoids a choice of representation:
h 𝜆 i = h 𝑠 | 𝐿ˆ | 𝑠 i (19.5)
Using (19.5) and assuming at the outset that the operator 𝐿ˆ is indepen- The Variation of the Mean
dent of time, let us write Value of a Quantity with Time
𝑑 𝑑
h 𝜆i = h 𝑠 (𝑡) | 𝐿ˆ | 𝑠 (𝑡) i
𝑑𝑡 𝑑𝑡
(19.6)
𝑑 ˆ 𝑑 | 𝑠 (𝑡) i
= h 𝑠 (𝑡) | |𝐿ˆ | 𝑠 (𝑡) i + h 𝑠 (𝑡) | 𝐿|
𝑑𝑡 𝑑𝑡
𝑑
−𝑖ℏ h 𝑠 (𝑡) | = h 𝑠 (𝑡) | 𝐻ˆ †
𝑑𝑡
or, taking into account the hermiticity of the Hamiltonian,
𝑑
−𝑖ℏ h 𝑠 (𝑡) | = h 𝑠 (𝑡) | 𝐻ˆ (19.7)
𝑑𝑡
Substituting (18.12) and (19.7) into (19.6), we get
𝑑 𝑖
h 𝜆 i = h 𝑠 (𝑡) | 𝐻ˆ 𝐿ˆ − 𝐿ˆ 𝐻ˆ | 𝑠 (𝑡) i
𝑑𝑡 ℏ
or
𝑑 𝑖
h 𝜆 i = h 𝑠 | 𝐻, ˆ 𝐿ˆ | 𝑠 i (19.8)
𝑑𝑡 ℏ
𝑑
If the quantity 𝑙 is an integral of motion, h 𝜆 i = 0. It follows from
𝑑𝑡
(19.8) that the above-mentioned condition 𝐻, ˆ 𝐿ˆ = 0 is a condition for the
li n e ar o p e r ato r s i n q ua n t u m m e c h a n i c s 181
𝑑
h 𝑠 | 𝐿¤̂ | 𝑠 i = h𝜆 i (19.9)
𝑑𝑡
Note that the result (19.4) could have been obtained unambiguously from the requirement of the
unitary invariance of the quantity h 𝜆 i using the expression (18.18) obtained for a special case. Indeed,
from the requirement of unitary invariance it follows that h 𝜆 i must be represented by an expression of
the type ∫
Φ∗ (𝑥) 𝐿ˆ𝑛 (𝑥) Φ(𝑥) 𝑑𝑥
and a comparison with the particular result (18.18) indicates that in this case 𝑛 must be put equal to 1.
182 b as ic c on c e pt s o f q ua n t u m m e ch a n ics
Note the importance of finding the form of the two basic “physical op-
erators”, the coordinate and the momentum of a microparticle. Knowing
these operators, we may obtain the energy operator [see (19.1)] and the
angular momentum operator [see (19.2)].
For simplicity we shall consider one-dimensional motion along the 𝑥- Operators of Coordinate and
axis (the result so obtained can be easily generalized to a three-dimensional Momentum
case). Taking into account the remarks made in Section 17 about the form
of Hermitian operator in its eigen representation [see (17.26)], we con-
clude that the operator of a coordinate in the coordinate representation is
the coordinate itself:
𝑥 (𝑥) = 𝑥 (20.1)
Let us now try to find the form of the momentum operator. First we shall
prove the following theorem. Suppose an operator 𝑂ˆ somehow transforms
a coordinate. If the Hamiltonian 𝐻ˆ remains invariant under this transfor-
mation, the operators 𝑂ˆ and 𝐻ˆ commute.
Proof : Let 𝑂ˆ 𝑥 = 𝑥 0; we act on 𝐻ˆ (𝑥) 𝜓 (𝑥) with the operator 𝑂ˆ to get
𝑂ˆ 𝐻ˆ (𝑥) 𝜓 (𝑥) = 𝐻ˆ (𝑥 0) 𝜓 (𝑥 0)
= 𝐻ˆ (𝑥) 𝜓 (𝑥 0)
= 𝐻ˆ (𝑥) 𝑂ˆ 𝜓 (𝑥),
Thus
𝑑
𝑂ˆ = 1 + Δ𝑥
𝑑𝑥
Proceeding from the properties of homogeneity of space, we conclude
that an operation by 𝑂ˆ must leave the Hamiltonian of a microparticle
li n e ar o p e r ato r s i n q ua n t u m m e c h a n i c s 183
But it was shown above in Section 19 that a commutation with the Hamil-
tonian expresses the law of conservation of a physical quantity. This
𝑑
means that is the operator of some physical quantity which is con-
𝑑𝑥
served. We know that momentum is a quantity whose conservation is a
consequence of the homogeneity of space (see Section 1). Consequently,
𝑑
the operator must coincide with the momentum operator of a mi-
𝑑𝑥
croparticle up to some constant factor:
𝑑
= 𝛾 𝑝ˆ𝑥 (20.3)
𝑑𝑥
The factor 𝛾 is determined from a consideration of the limiting transition
from quantum mechanics to classical mechanics (see Appendix B). It
𝑖
can be shown that 𝛾 = . Thus the operator of the 𝑥-component of a
ℏ
microparticle momentum in the coordinate representation has the form:
𝑑
𝑝ˆ𝑥 = −𝑖ℏ (20.4)
𝑑𝑥
The results (20.1) and (20.4) can be easily generalized to a three-dimensional
case:
𝑟®ˆ = 𝑟® (20.5)
𝑝®ˆ = −𝑖ℏ∇ (20.6)
Using (20.4), we can write an equation for the eigenfunctions of the 𝑥- Eigenfunctions of Momentum
component of the momentum:
𝑑
−𝑖ℏ 𝜓𝑝 (𝑥) = 𝑝𝑥 𝜓𝑝𝑥 (𝑥) (20.7)
𝑑𝑥 𝑥
It can be easily seen that (20.7) can be solved for any values of the param-
eter 𝑝𝑥 . Consequently, the momentum of a microparticle is not quantized
(the spectrum of the eigenvalues of the momentum is continuous).
1
Comparing the last two equations, we get 𝐴2 = . Consequently,
2𝜋ℏ
1 𝑖𝑝𝑥 𝑥
𝜓𝑝𝑥 (𝑥) = √ exp (20.9)
2𝜋ℏ ℏ
Note that the eigenfunction of the momentum (20.10) coincides with the
wave function (15.15) derived in Section 15 for a freely moving microparti-
cle.
Let us consider the equation (18.23) for eigenfunctions of a Hamilto- Schrödinger Equation
nian:
𝐻ˆ 𝜑 𝐸 (𝑥) = 𝐸 𝜑 𝐸 (𝑥) (20.11)
ℏ2 𝑑 2
𝐻ˆ (𝑥) = − + 𝑈 (𝑥) (20.12)
2𝑚 𝑑𝑥 2
Substituting (20.12) into (20.11) we get
ℏ2 𝑑 2
− 𝜑 𝐸 (𝑥) + [𝑈 (𝑥) − 𝐸] 𝜑 𝐸 (𝑥) = 0 (20.13)
2𝑚 𝑑𝑥 2
This is the one-dimensional Schrödinger equation. Generalizing it for the
three-dimensional case, we write
ℏ2
− 𝑟 ) + [𝑈 (®
Δ 𝜑 𝐸 (® 𝑟 ) − 𝐸] 𝜑 𝐸 (®
𝑟) = 0 (20.14)
2𝑚
Knowing the functions 𝜑 𝐸 (𝑥), we may write the expressions for ampli-
tudes of stationary states Ψ𝐸 (𝑥, 𝑡), since the time dependence in this case
has the universal form discussed in Section 13. Using (13.4) and taking
into account the fact that
we get
−𝑖𝐸𝑡
Ψ𝐸 (𝑥, 𝑡) = 𝜑 𝐸 (𝑥) exp (20.16)
ℏ
li n e ar o p e r ato r s i n q ua n t u m m e c h a n i c s 185
It can be easily seen that the functions Ψ𝐸 (𝑥, 𝑡) are the solutions of equa-
tion (18.22), where the Hamiltonian (20.12) has been used in place of 𝐻ˆ . In
this case, the equation has the form
𝜕 ℏ2 𝜕 2
𝑖ℏ Ψ=− Ψ + 𝑈 (𝑥) Ψ (20.17)
𝜕𝑡 2𝑚 𝜕𝑥 2
It is also called the Schrödinger equation More precisely, equation (20.13)
is called the time-independent Schrödinger equation and the equation
(20.17) is called the time-dependent Schrödinger equation.
By using (19.2) and (20.4) we can easily get the expressions for the Operators of the Angular Mo-
operators of projections of the angular momentum: mentum Projections and the
Square of Angular Momentum
𝜕 𝜕
𝑀𝑥 = −𝑖ℏ 𝑦
ˆ −𝑧
𝜕𝑧 𝜕𝑦
𝜕 𝜕
𝑀𝑦 = −𝑖ℏ 𝑧
ˆ −𝑥 (20.18)
𝜕𝑥 𝜕𝑧
𝜕 𝜕
𝑀𝑧 = −𝑖ℏ 𝑥
ˆ −𝑦
𝜕𝑦 𝜕𝑥
𝑥 = 𝑟 sin 𝜃 cos 𝜑
𝑦 = 𝑟 sin 𝜃 cos 𝜑 (20.20)
𝑧 = 𝑟 cos 𝜃
𝜕
Using (20.20), we represent the derivative 𝜓 in the form
𝜕𝜑
𝜕 𝜕𝜓 𝜕𝑥 𝜕𝜓 𝜕𝑦 𝜕𝜓 𝜕𝑧
𝜓= + +
𝜕𝜑 𝜕𝑥 𝜕𝜑 𝜕𝑦 𝜕𝜑 𝜕𝑧 𝜕𝜑
186 b as ic c on c e pt s o f q ua n t u m m e ch a n ics
𝜕𝜓 𝜕𝜓
= − 𝑟 sin 𝜃 sin 𝜑 + 𝑟 sin 𝜃 cos 𝜑
𝜕𝑥 𝜕𝑦
𝜕 𝜕
= 𝑥 −𝑦 𝜓
𝜕𝑦 𝜕𝑥
𝜕
In a similar way from the derivative 𝜓 we find
𝜕𝜃
𝜕 𝜕
𝑀𝑥 ± 𝑖 𝑀𝑦 = ℏ(exp{±𝑖𝜑 }) ± + 𝑖 cot 𝜃
ˆ ˆ (20.22)
𝜕𝜃 𝜕𝜑
These relations form the commutation rules for the operators of coordi- Commutation Relations
nate, momentum and angular momentum of a microparticle. Denoting
the Cartesian components of these operators by subscripts 𝑖, 𝑗, 𝑘, we can
write these commutation rules (it will be shown later on how they may be
derived):
Results (20.28) and (20.29) mean that components of the angular mo-
mentum and coordinate (angular momentum and momentum) with like
subscripts can be measured simultaneously, while those with different
subscripts cannot be measured simultaneously. These results also mean
that projections of the angular momentum cannot belong to complete sets
which include coordinates or the momentum components.
By using relations (20.19) and (20.30), we can establish one more rule
188 b as ic c on c e pt s o f q ua n t u m m e ch a n ics
[𝑀ˆ 2, 𝑀ˆ 𝑖 ] = 0 (20.31)
This means that we must include the square of the angular momentum
and anyone of the projections of the angular momentum in the same
complete set of quantifies.
Note that the simultaneous measurability of all the components of momentum and the impossibility
of similar measurement for the angular momentum components have a very simple explanation. The
fact is that the parallel translations associated with the momentum operator are commutative, while
the rotations associated with the angular momentum operator are non-commutative. It is immaterial
whether we move first along the 𝑥-axis and then along the 𝑦-axis, or in the reverse order.
However, the sequence of rotations is certainly not immaterial. Take, for example, a point on the
𝑧-axis and make two successive rotations through 90° - in one case first around the 𝑥-axis and then
around the 𝑧-axis, in the other ease first around the 𝑧-axis and then around the 𝑧-axis. It can be easily
seen that the final positions of the point are different in these two cases.
The inversion operator 𝑃ˆ is defined in the following way: The Inversion Operator; Parity
𝑃𝜓
ˆ (®
𝑟 , 𝑡) = 𝑃𝜓 (−®
𝑟 , 𝑡) (20.32)
Suppose [𝑃,
ˆ 𝐻ˆ ] = 0. In this case, according to (19.10), the parity is
a conservable quantity. If at the initial moment of time the microparti-
cle was, for example, in a state with even parity, it must have the same
parity at subsequent moments of time (which, of course, imposes certain
restrictions on the possible changes in the state of the micro-particle).
Let us write the equation for the eigenfunctions of the operator 𝑀ˆ𝑧 Eigenvalues and Eigenfunc-
defined by (20.21): tions Operators 𝑀ˆ𝑧 and 𝑀ˆ 2
𝜕
−𝑖ℏ 𝜓 = 𝑀𝑧𝜓 (20.34)
𝜕𝜑
The solutions of this equation are of the form of the
𝑖𝑀𝑧 𝜑
𝜓 (𝜑) = 𝐴 exp (20.35)
ℏ
The reader is already familiar with this results: the projection of angu-
lar momentum is quantized; it assumes values differing by multiples of
Planck’s constant (see Section 2). The factor 𝐴 in (20.35) is determined
from the normalization condition
∫2𝜋
∗
𝜓𝑚 𝜓𝑚 𝜑 = 1
0
𝑀 2 = ℏ2 𝑙 (𝑙 + 1), 𝑙 = 0, 1, 2, (20.39)
−𝑙 to 𝑙). The functions 𝑃𝑙|𝑚 | (cos 𝜃 ) appearing in (20.40) are essentially the
associated Legendre functions. We remind the reader that
𝑑𝑚
𝑃𝑙𝑚 (𝑥) = (1 − 𝑥 2 )𝑚/2 𝑃𝑙 (𝑥) (20.41)
𝑑𝑥 𝑚
∫2𝜋 ∫𝜋
∗
𝜓𝑙𝑚 (𝜃, 𝜑) 𝜓𝑙∗0𝑚0 (𝜃, 𝜑) sin 𝜃 𝑑𝜃 𝑑𝜑 = 𝛿𝑙𝑙 0 𝛿𝑚𝑚0 (20.43)
0 0
If the result (20.36) is known, we can derive the result (20.39) by assuming
that
𝑀 2 = 3 h 𝑀𝑧2 i = 3ℏ2 h 𝑚 2 i
Õ
𝑙
𝑚2 Õ𝑙
𝑚2 1
h𝑚 2 i = =2 = 𝑙 (𝑙 + 1)
2𝑙 + 1 𝑚=0
2𝑙 + 1 3
𝑚=−𝑙
It follows from (20.18) that the inversion operators commute with Parity and Angular Momen-
operators of any projection of the angular momentum. Moreover, the tum
inversion operators and the operators of the square of the angular momen-
tum commute:
[𝑃,
ˆ 𝑀ˆ 𝑖 ] = 0, ˆ 𝑀ˆ 2 ] = 0
[𝑃, (20.44)
This mean that the operators 𝑃ˆ and 𝑀ˆ 𝑖 have a common complete system
of eigenfunctions. The same applies to the operators 𝑃ˆ and 𝑀ˆ 2 . From this
it follows in particular that a state with a definite orbital quantum num-
ber 𝑙 must also be characterized by a definite spatial parity. In spherical
coordinates the inversion is of the form
𝑟 → 𝑟, 𝜃 → 𝜋 − 𝜃, 𝜑 → 𝜑 + 𝜋. (20.45)
It follows hence that states with even 𝑙 have an even parity while the
states with odd 𝑙 have an odd parity.
𝑀 2, 𝑀𝑧 and 𝑃 [see (3.7b). Note that the parity and the angular momen-
tum occur in the same complete set of quantities. Formally, this is a
consequence of relations (20.44). However, one can start from consid-
erations based on direct physical intuition. In fact, the obvious “affinity”
between the parity and the angular momentum is connected with the
above-mentioned fact that the inversion operation includes rotation in
addition to reflection, The order in which these operations are carried out
is immaterial; rotation can follow reflection or, the other way round, it
can precede reflection.
So far, we have several times used the fact that the apparatus of quan- The Relations of Classical
tum mechanics is based on the well-known equations of classical mechan- Mechanics in Operator Form
ics written, however, in operator form. This fact is so important that it is
appropriate to return to it once again.
𝑝ˆ𝑥
𝑥= (20.47)
𝑚
This is the well-known classical result: the velocity is equal to momentum
divided by mass. In quantum-mechanical interpretation this result means
that the velocity operator is equal to the momentum operator divided by
the mass. Further, we substitute the operator 𝐿ˆ = 𝑝ˆ𝑥 in (19.10). Using the
expression (20.12) for 𝐻ˆ as before, we get
𝑑
𝑝𝑥 = − 𝑈 (𝑥) (20.48)
𝑑𝑥
It can be easily seen that this is just Newton’s second law of motion,
written in operator form.
Remember, that the operators of the type 𝑥¤̂ and 𝑝¤̂𝑥 are introduced in
accordance with the definition (19.9). Hence, it follows that the results
(20.47) and (20.48) indicate the validity of the classical relations for the
mean values of physical quantities:
𝑑
h 𝑥 i = h 𝑠 | 𝑝ˆ𝑥 | 𝑠 i /𝑚 (20.47a)
𝑑𝑡
𝑑 𝑑
h 𝑝𝑥 i = − 𝑈 (𝑥) (20.48a)
𝑑𝑡 𝑑𝑥
Ehrenfest was the first to point out this and hence relations of the type
(20.47a) or (20.48a) are called Ehrenfest theorems. In short, Ehrenfest’s
192 b as ic c on c e pt s o f q ua n t u m m e ch a n ics
theorems state that classical relations for physical quantities are trans-
formed in quantum mechanics into relations for mean values of physical
quantities.
We shall mention three types of problems involving the solution of the Some Characteristic Problems
Schrödinger equation. in Quantum Mechanics
Examples of problems of first two types are given in this section and
in Section 24. The third type of problems can be found in Section 25. 6 de Ter Haar, editor (1965). Selected
Naturally, we shall limit ourselves to just a few typical examples. But it Problems in Quantum Mechanics. Aca-
demic Press, N.Y; and Flügge, S. (1971).
should be mentioned here that the applied aspects of quantum mechanics Practical Quantum Meehanics, Vols. 1 and 2.
are reflected quite comprehensively in the existing literature (in this Springer, Berlin
𝑑 2𝜑
+ 𝑘 2𝜑 = 0 (21.2)
𝑑𝑥 2
where
𝑘 2 = 2𝑚𝐸/ℏ2 (21.3)
𝑘𝑎 = 𝜋𝑛 (21.7)
194 b as ic c on c e pt s o f q ua n t u m m e ch a n ics
where 𝑛 is an integer.
Taking (21.3) into account, the last result can be rewritten in the follow-
ing form:
𝑛 2 𝜋 2 ℏ2
𝐸𝑛 = (21.8)
2𝑚𝑎 2
The expression (21.8) determines the spectrum of values of the energy
(energy levels) of the particle in the potential well. It coincides with the
expression (5.2) derived earlier.
Let us consider a rectangular potential well shown in Figure 21.1 (a). Rectangular Potential Well
Since the particle is inside the well, 𝐸 < 𝑈 1 and 𝐸 < 𝑈 2 . The rectangularity with “Walls” of Finite Height
of the potential enables us to clearly distinguish three spatial regions:
region 1 (𝑥 < 0), region 2 (0 ⩽ 𝑥 ⩽ 𝑎) and region 3: (𝑥 > 𝑎).
We shall consider these regions separately and will then combine these
results at the boundaries of the regions, i.e. at the points 𝑥 = 0 and 𝑥 = 𝑎.
The Schrödinger equation (20.13) has the following form:
𝑑 2𝜑
for region 1: − 𝜘12𝜑 = 0, where 𝜘12 = 2𝑚(𝑈 1 − 𝐸)/ℏ2 (21.10a)
𝑑𝑥 2
𝑑 2𝜑
for region 2: + 𝑘 2𝜑 = 0, where 𝑘 2 = 2𝑚𝐸/ℏ2, and (21.10b)
𝑑𝑥 2
𝑑 2𝜑
for region 3: − 𝜘22𝜑 = 0, where 𝜘22 = 2𝑚(𝑈 2 − 𝐸)/ℏ2 (21.10c)
𝑑𝑥 2
The general solutions of these differential equations may be written in the
following form:
Figure 21.1: A particle with energy 𝐸 in a
rectangular potential well with “walls” of
for region 1: 𝜑 1 = 𝐴1 exp(𝜘1 𝑥) + 𝐵 1 exp(−𝜘1 𝑥) (21.11a) finite height, 𝑈 1 and 𝑈 2 .
for region 2: 𝜑 2 = 𝐴2 exp(𝑖 𝑘 𝑥) + 𝐵 2 exp(−𝑖 𝑘 𝑥), and (21.11b)
for region 3: 𝜑 3 = 𝐴3 exp(𝜘2 𝑥) + 𝐵 3 exp(−𝜘2 𝑥) (21.11c)
li n e ar o p e r ato r s i n q ua n t u m m e c h a n i c s 195
[Note that the solution of equation (21.10b) may be written in the form
(21.11b) or (21.5)] The boundedness of the wave function requires that 𝐵 1
and 𝐴3 be put equal to zero. Thus,
𝜑 1 = 𝐴1 exp(𝜘1 𝑥)
𝜑 2 = 𝐴2 exp(𝑖 𝑘 𝑥) + 𝐵 2 exp(−𝑖 𝑘 𝑥) (21.12)
𝜑 3 = 𝐵 3 exp(−𝜘2 𝑥)
∫𝑎+Δ
𝑑𝜑 𝑑𝜑 2𝑚
(𝑎 + Δ) − (𝑎 − Δ) = 2 [𝑈 (𝑥) − 𝐸]𝜑 𝑑𝑥
𝑑𝑥 𝑑𝑥 ℏ
𝑎−Δ
Since the functions under the integral sign are bounded, in the limit of
Δ → 0 this integral vanishes. As a result we get
𝑑𝜑 𝑑𝜑
(𝑎 + 0) = (𝑎 − 0)
𝑑𝑥 𝑑𝑥
which was to be proved. Returning to our problem, we write the bound-
ary conditions (conditions of ‘piecing’ together the solutions at the bound-
aries of the region):
𝜑 1 (0) = 𝜑 2 (0)
𝜑 2 (𝑎) = 𝜑 3 (𝑎)
𝑑𝜑 1 𝑑𝜑 2 (21.13)
(0) = (0)
𝑑𝑥 𝑑𝑥
𝑑𝜑 2 𝑑𝜑 3
(𝑎) = (𝑎)
𝑑𝑥 𝑑𝑥
Substituting the expressions (21.12) in these equations, we get the system
of equations for the coefficients (𝐴1, 𝐴2, 𝐵 2, 𝐵 3 ):
𝐴1 = 𝐴2 + 𝐵 2 ,
𝐴2 exp(𝑖𝑘𝑎) + 𝐵 2 exp(−𝑖𝑘𝑎) = 𝐵 3 exp(−𝜘2 𝑎),
(21.14)
𝜘1 𝐴1 = 𝑖𝑘 (𝐴2 − 𝐵 2 )
𝑖𝑘𝐴2 exp(𝑖𝑘𝑎) − 𝑖𝑘𝐵 2 exp(−𝑖𝑘𝑎) = −𝜘2 𝐵 3 exp(−𝜘2 𝑎)
that such a system should have non- trivial solutions, it is necessary that
its determinant should be equal to zero. Equating the determinant of the
system to zero, we get an equation for the energy 𝐸 (we recall that the
quantities 𝑘, 𝜘1, 𝜘2 are expressed in terms of 𝐸). The solutions of this
equation will give us the possible values of the energy of the particle.
𝜑 2 = 𝐶 sin(𝑘𝑥 + 𝑏)
(this form is equivalent to the one used earlier; the reader may indepen-
dently express the coefficients 𝐶 and 𝑏 in terms of the old coefficients 𝐴2
and 𝐵 2 ). The system (21.14) will now have the following form:
𝐴1 = 𝐶 sin 𝑏
𝐶 sin(𝑘𝑎 + 𝑏) = 𝐵 3 exp(−𝜘2𝑎)
(21.15)
𝜘1𝐴1 = 𝑘𝐶 cos 𝑏
𝑘𝐶 cos(𝑘𝑎 + 𝑏) = −𝜘2 𝐵 3 exp(−𝜘2𝑎)
Dividing the third equation of this system by first and the fourth by sec-
ond, we get the fourth by second, we get
𝜘1 = 𝑘 cot 𝑏,
(21.16)
𝜘2 = −𝑘 cot(𝑘𝑎 + 𝑏)
We now have a system of two equations in place of the system with four
equations. From the first equation of the system (21.16) we find cot 𝑏 =
𝜘1 /𝑘, hence
𝑘ℏ
sin 𝑏 = (1 + cot2 𝑏) −1/2 = √
2𝑚𝑈 1
li n e ar o p e r ato r s i n q ua n t u m m e c h a n i c s 197
𝑘ℏ
sin(𝑘𝑎 + 𝑏) = − √
2𝑚𝑈 2
𝑘ℏ 𝑘ℏ
𝑘𝑎 = 𝑛𝜋 − arcsin √ − arcsin √ (21.17)
2𝑚𝑈 1 2𝑚𝑈 2
where 𝑛 is an integer. Figure 21.2 shows the left- and right-hand sides
of equation (21.17) as a function of 𝑘.
Figure 21.2: The number of energy levels
of the particle are determined by the
In the situation shown in the figure, the particle has three energy levels intersection of the straight line 𝑦 = 𝑘𝑎
corresponding to values of 𝑘 equal to 𝑘 1, 𝑘 2, 𝑘 3 . If we change the width of with values of 𝑘. The deeper the well,
there will be more intersections leading to
the well, the slope of the straight line 𝑦 = 𝑘𝑎 will change thus changing more levels.
the the position and the number of possible energy levels.
A decrease in the width of the well will decrease the slope of the
straight line. The energy levels will “creep” out of the well, and their
number will gradually decrease. An increase in the width of the well will
raise the line 𝑦 = 𝑘𝑎; it will intersect a larger number of branches of the
inverse sine graph, thus leading to a larger number of levels in the well.
As 𝑎 → ∞, the number of levels in the well will rise indefinitely, and
we finally get a continuous energy spectrum. It is easy to examine simi-
larly the effect of a change in the depth of tho well on the spectrum: the
greater is the depth, the more levels there are in the well.
We should proceed further in the following way: find from (21.17) the
possible values of 𝑘 and the values 𝐸, 𝜘1 and 𝜘2 corresponding to them,
then substitute these values in (21.15) and solve the system of equations
containing the coefficients, and then substitute the final result in the
expression (21.12) for the wave functions. However, on account of the
mathematical complications involved, we shall not embark on this venture
here.
where
1 𝜕2 1 𝜕 𝜕
Δ𝜃 𝜑 = + sin 𝜃 (21.19)
sin2 𝜃 𝜕𝜑 2 sin 𝜃 𝜕𝜃 𝜕𝜃
Taking this into account, we can rewrite the Schrödinger equation (20.14)
in the form
1 𝜕 2 𝜕 1 2𝑚
𝑟 𝜓 + 2 Δ𝜃 𝜑 𝜓 (𝑟, 𝜃, 𝜑) + 2 [𝐸 − 𝑈 (𝑟 )] 𝜓 (𝑟, 𝜃, 𝜑) = 0 (21.20)
𝑟 2 𝜕𝑟 𝜕𝑟 𝑟 ℏ
Equation (21.20) allows the separation of variables. This means that its
solution may be found in the form of a product of two functions, one of
which depends only on 𝑟 , and the other on the angular coordinates 𝜃 and
𝜑:
𝜓 (𝑟, 𝜃, 𝜑) = 𝑅(𝑟 ) Φ(𝜃, 𝜑) (21.21)
Since the left- and right-hand sides of (21.22) depend on different inde-
pendent variables (on 𝑟 and on 𝜃 and 𝜑, respectively), both sides must be
equal to some constant, which we denote by 𝜆. Introducing this constant,
we write
−Δ𝜃 𝜑 Φ(𝜃, 𝜑) = 𝜆 Φ(𝜃, 𝜑) (21.23)
𝜆 = 𝑙 (𝑙 + 1), 𝑙 = 0, 1, 2, (21.24)
s
2𝑙 + 1 (𝑙 − |𝑚|)! |𝑚 |
Φ(𝜃, 𝜑) = 𝑃 cos 𝜃 exp(𝑖𝑚𝜑) ≡ 𝑌𝑙𝑚 (𝜃, 𝜑) (21.25)
4𝜋 (𝑙 + |𝑚|)! 𝑙
r
15
𝑌2, ±2 = sin2 𝜃 exp(±2𝑖𝜑) (21.26c)
32𝜋
We emphasize that the “angular part” of the wave function is independent
of the particular form of the potential 𝑈 (𝑟 ); this is a direct and important
consequence of the spherical symmetry of the potential.
We now turn to the “radial part” of the wave function, i.e. to the func-
tion 𝑅(𝑟 ). According to (21.22) and (21.24), this must be a solution of the
equation
1 𝑑 2 𝑑𝑅 2𝑚
𝑟 + 2 [𝐸 − 𝑈𝑙 (𝑟 )] 𝑅(𝑟 ) = 0 (21.27)
𝑟 𝑑𝑟
2 𝑑𝑟 ℏ
where we have introduced the notation
ℏ2𝑙 (𝑙 + 1)
𝑈𝑙 (𝑟 ) = 𝑈 (𝑟 ) + (21.28)
2𝑚𝑟 2
It should be noted that (21.27) may be reduced to the one-dimensional
Schrödinger equation with a special boundary condition at 𝑟 = 0. For this
we must use the substitution
𝜑 (𝑟 ) = 𝑟𝑅(𝑟 ) (21.29)
and in view of the boundedness of the function 𝑅(𝑟 ), require that the
condition 𝜑 (0) = 0 be satisfied. It can be easily seen that the substitution
(21.29) in fact converts (21.27) into the one-dimensional Schrödinger
equation
𝑑 2𝜑 2𝑚
+ 2 [𝐸 − 𝑈𝑙 (𝑟 )] 𝜑 (𝑟 ) = 0 (21.30)
𝑑𝑟 2 ℏ
In this case the boundary condition 𝜑 (0) = 0 corresponds to the one-
dimensional potential well having an infinitely high vertical wall on the
left (at 𝑟 = 0).
We shall establish a formal analogy between the time-dependent The Continuity Equation and
Schrödinger equation and the continuity equation, which is widely used the Schrödinger Equation
in classical physics, especially in hydrodynamics. We assume that there is
a certain medium (for example, a liquid) described by the functions 𝜌 (®
𝑟)
and 𝑣® (®
𝑟 ). [𝜌 (®
𝑟 ) is the density of the medium and 𝑣® (®
𝑟 ) is the velocity of
the particles of the medium at the point 𝑟®; naturally, these functions may
also depend on time]. Let us imagine a certain volume 𝑉 in the medium
to be isolated. The change in the quantity of liquid in this volume per unit
200 basi c c o n c e p t s o f q ua n t u m m e c h a ni c s
time is equal to ∫
𝜕
𝜌 𝑑𝑉
𝜕𝑡
𝑉
Thus, ∫ ∮
𝜕
𝜌 𝑑𝑉 + 𝜌 𝑣® 𝑑 𝑆® = 0
𝜕𝑡
𝑉 𝑆
𝑑Ψ 𝑑Ψ∗
Substituting into this and from the Schrödinger equation (21.33),
𝑑𝑡 𝑑𝑡
and from the complex conjugate of equation (21.33), we get
∫ ∫
𝜕 𝑖ℏ
ΨΨ∗𝑑𝑉 = (Ψ∗ ΔΨ − ΨΔΨ∗ ) 𝑑𝑉
𝜕𝑡 2𝑚
𝑉 𝑉
∫
𝑖ℏ
= div (Ψ∗ ∇Ψ − Ψ∇Ψ∗ ) 𝑑𝑉
2𝑚
𝑉
𝑖ℏ
𝑗® = (Ψ∇Ψ∗ − Ψ∗ ∇Ψ)
2𝑚 (21.37)
𝑖ℏ
= (𝜑∇𝜑 ∗ − 𝜑 ∗ ∇𝜑)
2𝑚
If we interpret (21.36) as the density of particles, then the vector (21.37)
may be considered as the vector of density of the flow of particles. With
such an interpretation the quantum-mechanical continuity equation
(21.35) expresses the law of conservation of the number of particles.
Let us consider a one-dimensional rectangular potential barrier (Fig- Passage of a Particle Under or
ure 21.3) and assume that particles arrive at it from the left with an energy Over a Potential Barrier
𝐸 which is less than the height 𝑈 of the barrier. We can isolate three spa-
tial regions and write the solutions of the Schrödinger equation (20.13) for
these regions:
√
2𝑚𝐸
𝜑 1 (𝑥) = 𝐴1 exp(𝑖𝑘𝑥) + 𝐵 1 exp(−𝑖𝑘𝑥); 𝑘=
ℏ
p
2𝑚(𝑈 − 𝐸) (21.39)
𝜑 2 (𝑥) = 𝐴2 exp(𝜘𝑥) + 𝐵 2 exp(−𝜘𝑥); 𝜘=
ℏ
𝜑 3 (𝑥) = 𝐴3 exp(𝑖𝑘𝑥) + 𝐵 3 exp(−𝑖𝑘𝑥)
the particles are moving in the positive direction, we must exclude the
second term in the function 𝜑 3 : 𝐵 3 = 0. The other coefficients are non-
zero. The term with 𝐴1 describes particles falling on the barrier, the term
with 𝐵 1 describes those reflected from the barrier, while the term with 𝐴3
describes particles which have passed through the barrier.
𝐴1 + 𝐵 1 = 𝐴2 + 𝐵 2
𝐴2 exp(𝜘𝑎) + 𝐵 2 exp(−𝜘𝑎) = 𝐴3 exp(𝑖𝑘𝑎)
(21.40)
𝑖𝑘 (𝐴1 − 𝐵 1 ) = 𝜘(𝐴2 − 𝐵 2 )
𝜘 [𝐴2 exp(𝜘𝑎) − 𝐵 2 exp(−𝜘𝑎)] = 𝑖𝑘 𝐴3 exp(𝑖𝑘𝑎)
It turns out that we have just four equations for five coefficients! But
actually, only four and not five coefficients are known. The density of
flow of particles incident on the barrier (𝑗inc ) must be given. This density
is given by (21.38), where we must substitute 𝜑 = 𝐴1 exp(𝑖𝑘𝑥). As a result
of this, we get
𝑗inc = |𝐴1 | 2 ℏ𝑘/𝑚 (21.41)
and for the density of flow of particles passing through the barrier we
li n e ar o p e r ato r s i n q ua n t u m m e c h a n i c s 203
have
𝑗tr = |𝐴3 | 2 ℏ𝑘/𝑚 (21.43)
Usually in such problems the density 𝑗inc is chosen in such a way that
𝐴1 = 1. In this case the system (21.40) assumes the form
1 + 𝐵 1 = 𝐴2 + 𝐵 2
𝐴2 exp(𝜘𝑎) + 𝐵 2 exp(−𝜘𝑎) = 𝐴3 exp(𝑖𝑘𝑎)
(21.44)
𝑘 (1 − 𝐵 1 ) = 𝜘 (𝐴2 − 𝐵 2 )
𝜘 [𝐴2 exp(𝜘𝑎) − 𝐵 2 exp(−𝜘𝑎)] = 𝑖𝑘 𝐴3 exp(𝑖𝑘𝑎)
We can determine the fraction of particles that has passed through the
barrier:
𝑗tr
𝐷= (21.45)
𝑗inc
The quantity 𝐷 is called the transmission coefficient of the barrier. Solving
the system (21.44) (we shall omit the steps), we get
" 2 2#
4𝑖𝑘 𝑖𝑘 𝑖𝑘
𝐴3 = − exp(−𝑖𝑘𝑎) exp(𝜘𝑎) 1 − − exp(−𝜘𝑎) 1 + (21.46)
𝜘 𝜘 𝜘
4𝑘 2 𝜘 2
𝐷= (21.47)
4𝑘 2 𝜘 2 + (𝑘 2 + 𝜘 2 ) 2 sinh2 (𝜘𝑎)
where
𝐸 𝐸
𝐷 0 = 16 1−
𝑈 𝑈
In addition to the transmission coefficient, we also have the coefficient of
reflection at the barrier, defined as the fraction of the particles reflected by
the barrier: 𝑅 = 𝑗ref /𝑗inc . It is clear from basic principles that 𝐷 + 𝑅 = 1 (all
the particles not passing through the barrier must be reflected by it).
Finally we consider the case when a particle passes over the barrier
204 basi c c o n c e p t s o f q ua n t u m m e c h a ni c s
(𝑘 2 − 𝐾 2 ) 2 sin2 (𝐾𝑎)
𝑅= (21.50)
4𝑘 2 𝐾 2 + (𝑘 2 − 𝐾 2 ) 2 sin2 (𝐾𝑎)
Using (21.47) and (21.50) we can find the dependence of the transmis-
sion coefficient 𝐷 on the ratio 𝐸/𝑈 . This dependence is shown graphically
in Figure 21.4. The same figure shows the dependence 𝐷 (𝐸/𝑈 ) for a Figure 21.4: Classical and quantum
classical particle (dotted line). A comparison of the solid curve with the transmission and reflection coefficients.
The Hamiltonian is of the form [See Appendices E and F]. Linear Harmonic Oscillator
ℏ2 𝑑 2 𝑚𝜔 2𝑥 2
𝐻ˆ = − + (22.1)
2𝑚 𝑑𝑥 2 2
It is obtained from (4.5) by taking (19.1) and (20.4) into account. The
eigenvalues are
1
𝐸𝑛 = ℏ𝜔 𝑛 + 𝑛 = 0, 1, 2, (22.2)
2
[for 𝑛 = 0 we get from (22.2) the energy of zero-point oscillations, which
was determined in Section 4 on the basis of the uncertainty relations].
The eigenfunctions are
p
𝜑𝑛 (𝑥) = 4 𝑚𝜔/ℏ exp −𝜉 2 /2 𝐻𝑛 (𝜉) (22.3)
li n e ar o p e r ato r s i n q ua n t u m m e c h a n i c s 205
p
where 𝜉 = 𝑥 𝑚𝜔/ℏ, and 𝐻𝑛 (𝜉) are called Hermite polynomials. Let us
write down the expressions for the first few functions 𝜑𝑛 (𝑥):
√
𝜑 0 (𝑥) = (𝑥 0 𝜋) −1/2 exp −𝑥 2 /2𝑥 02 (22.4a)
√
𝜑 1 (𝑥) = (2𝑥 0 𝜋) −1/2 exp −𝑥 2 /2𝑥 02 2𝑥/𝑥 0 (22.4b)
√ 𝑥2
𝜑 2 (𝑥) = (8𝑥 0 𝜋) −1/2 exp −𝑥 2 /2𝑥 02 4 2 − 2 (22.4c)
𝑥0
p
where 𝑥 0 = ℏ/𝑚𝜔.
The problem of the hydrogen atom is a well-known example of the The Hydrogen Atom
motion of an electron in a spherically symmetrical Coulomb field. The
Hamiltonian has the form
ℏ2 𝑒2
𝐻ˆ = − Δ − (22.5)
2𝑚 𝑟
[it is obtained from (4.1) by using (19.1) and (20.6)]. The eigenvalues of
this Hamiltonian are given by the following familiar expression [see (2.5a)-
(2.5b)]:
𝑚𝑒 4
𝐸𝑛 = − , 𝑛 = 1, 2, 3. (22.6)
2ℏ2𝑛 2
The eigenfunctions of the Hamiltonian (22.5) may be expressed in the
form
Here 𝑌𝑙𝑚 (𝜃, 𝜑) are spherical functions. They define the “angular part” of
the wave function irrespective of the particular form of the spherically
symmetrical potential; 𝑅𝑛𝑙 (𝑟 ) is the “radial part” of the wave function, it is
defined by (21.30) with the Coulomb potential 𝑈 (𝑟 ) = −𝑒 2 /𝑟 . The form of
the function 𝑅𝑛𝑙 (𝑟 ) is described by the expression
𝑙
𝑟 2𝑟 2𝑟
𝑅𝑛𝑙 (𝑟 ) = const exp 𝐿𝑛+1
2𝑙+1
(22.8)
𝑟 1𝑛 𝑟 1𝑛 𝑟 1𝑛
in Section 21 [see (21.26). We shall now give the expressions for the first
few functions 𝑅𝑛𝑙 (𝑟 ):
By using (22.9) and (21.26), we can write the first few eigenfunctions of
the Hamiltonian (22.5):
The function (22.10a) describes the ground state of the hydrogen atom,
while the functions (22.10b)-(22.10d) describe the excited states corre-
sponding to the first excited energy level (𝑛 = 2).
It follows from (22.6) that the energy of an electron in the hydrogen On Degeneracy of Energy
atom is determined only by the quantum number 𝑛, while the states (the Levels
functions 𝜓𝑛𝑙𝑚 ) are determined by three quantum numbers 𝑛, 𝑙 and 𝑚.
Besides, when considering electronic states we must take into account
the quantum number 𝜎 which does not occur in these expressions. Since
for a given value of the principal quantum number 𝑛 the orbital quantum
number 𝑙 assumes integral values from 0 to 𝑛 − 1, and for every 𝑙 the
magnetic quantum number acquires 2𝑙 + 1 values, the following 𝑔𝑛 states
must correspond to an energy level 𝐸𝑛 :
Õ
𝑛−1
𝑔𝑛 = 2 (2𝑙 + 1) = 2𝑛 2 (22.11)
𝑙=0
(the factor 2 takes account of the two spin states of the electron). This
means that the eigenvalue 𝐸𝑛 of the Hamiltonian (22.5) (in other words,
the nth energy level) is 2𝑛 2 -fold degenerate.
the atomic system. Thus, for example, owing to the spherical symmetry
of intra-atomic fields, there is degeneracy of the quantum numbers 𝑚 and
𝜎 – the energy is independent of the orientation of the orbital momentum
and the spin momentum of the electron. The degeneracy of the quantum
number 𝑙 is associated with the specific nature of the Coulomb potential;
in non-Coulomb fields the energy of the electron depends not only on 𝑛
but also on 𝑙.
The Hamiltonian of a crystal consisting of 𝑁 nuclei and 𝑍 𝑁 electrons Crystal; the Adiabatic Approxi-
can be written in the form mation
1 Õ ˆ2 1 Õ ˆ2
𝑁 𝑍𝑁
𝐻ˆ = 𝑃𝑖 + 𝑝𝑘 + 𝑈 1 ({𝑟®𝑘 }) + 𝑈 2 ({𝑅®𝑖 }) + 𝑈 3 ({𝑟®𝑘 }, {𝑅®𝑖 }) (22.12)
2𝑀 𝑖 2𝑚
𝑘
where 𝑀 is the mass of the nucleus, 𝑃ˆ𝑖 is the momentum operator for
the 𝑖th nucleus, 𝑚 is the mass of the electron, 𝑝𝑘 is the momentum oper-
ator for the 𝑘th electron, {𝑟®𝑘 } is the set of coordinates of the electrons,
{𝑅®𝑖 } is the set of coordinates of the nuclei. The function 𝑈 1 describes the
interaction of electrons. It is of the form
1 Õ Õ 𝑒2
𝑈1 = (22.13)
2 𝑟𝑘𝑙
𝑘≠𝑙
where 𝑟𝑘𝑙 is the distance between the 𝑘th and 𝑙th electrons. The function
𝑈 2 describes the mutual interaction of the nuclei, while the function 𝑈 3
describes the interaction of the nuclei with the electrons.11 11 The functions 𝑈 , 𝑈 , 𝑈 describing
1 2 3
the various interaction potentials, in fact
correspond to operators in the coordinate
Since 𝑀 𝑚, the nuclei move much more slowly than the electrons. representation.
This permits us to consider the motions of nuclei and electrons sepa-
rately: when considering the motion of electrons, we assume that the
nuclei are stationary, while when considering the motion of the nuclei,
we assume that the electrons collectively create an average field which
is independent of the coordinate of individual electrons. In this case the
wave function of the crystal may be represented in the form of a product
of “nuclear” and “electronic” functions:
1 Õ ˆ2
𝑁
𝐻ˆ1 = 𝑃 + 𝑈 2 ({𝑅®𝑖 }) (22.15)
2𝑀 𝑖 𝑖
1 Õ ˆ2 Õ Õ 𝑒 2
𝐻ˆ2 = 𝑝𝑘 + + 𝑈 3 ({𝑟®𝑘 }) (22.16)
2𝑚 𝑟𝑘𝑙
𝑘 𝑘≠𝑙
1 Õ ˆ2 1 Õ Õ 𝑒 2 Õ
𝐻ˆ = 𝑝𝑘 + + 𝑈 4 (𝑟®𝑘 ) (22.20)
2𝑚 2 𝑟𝑘𝑙
𝑘 𝑘≠𝑙 𝑘
li n e ar o p e r ato r s i n q ua n t u m m e c h a n i c s 209
1 Õ Õ 𝑒2
2 𝑟𝑘𝑙
𝑘 𝑙
1 Õ Õ 𝑒2 Õ
≈ 𝑈 5 (𝑟®𝑘 ) (22.21)
2 𝑟𝑘𝑙
𝑘≠𝑙 𝑘
®ˆ2
𝑝
𝑟 ) 𝜑 (®
2𝑚 + 𝑈 4 (®
𝑟 ) + 𝑈 5 (® 𝑟 ) = 𝐸 𝜑 (®
𝑟) (22.22)
Here 𝑝®ˆ and 𝑟® are the momentum operator and the coordinate of one of the
“collectivized” electrons, 𝐸 being the energy of the electron.
𝑈 (®
𝑟 ) = 𝑈 4 (®
𝑟 ) + 𝑈 5 (®
𝑟) (22.23)
The potential 𝑈 (®
𝑟 ) is a periodic function with the period of the crystal
lattice. It will be shown in Section 24 that the energy of an electron mov-
ing in a periodic field is broken up into alternate bands of allowed and
forbidden values, i.e. has a band structure. An electron hound to an atom
has energy levels, while a free electron is characterized by a continuous
energy spectrum. An electron “collectivized” by the crystal occupies an
“intermediate” position to a certain extent – it is “free”, but only within
the limits of the crystal. The band structure of the energy states of such
an electron is obvious and is “intermediate” between the structure of
discrete levels and that of a continuous spectrum.
We shall consider the system of a bound electron plus radiation. In the The Hamiltonian of the In-
absence of interaction between the electron and the radiation, the system teraction of an Electron with
is described by the “unperturbed” Hamiltonian: Electromagnetic Radiation
𝑝ˆ 2
𝐻ˆ0 = − + 𝑈 + 𝐻ˆ𝛾 (22.25)
2𝑚
𝑝ˆ 2
where + 𝑈 is the Hamiltonian of the electron, and 𝐻ˆ𝛾 is the Hamilto-
2𝑚
nian of the radiation. In the case of interaction between the electron and
the radiation, the system is described by a “perturbed” Hamiltonian:
𝑒 2
𝑝®ˆ − 𝐴®
𝐻ˆ = − 𝑐 (22.26)
2𝑚 + 𝑈 + 𝐻ˆ𝛾
where 𝐴® is the operator of the vector potential of the radiation field [we
®ˆ 𝑟 ) = 𝐴(®
recall that in the coordinate representation 𝐴(® ® 𝑟 ).12 12 It is shown in classical field theory
[for example, see Landau and Lifshitz, The
Classical Theory of Fields ] that the inter-
Note that the field potentials here have been chosen in such a way that action of a charge with an electromagnetic
the well-known calibration conditions div 𝐴® = 0 and 𝜑 = 0 (𝜑 is the scalar field may be considered by replacing 𝑝® by
𝑒 ®
𝑝® − 𝐴. We use this classical result here,
potential of the field) are satisfied. Next, we represent the Hamiltonian in 𝑐
replacing the dynamic variables with the
the following form: corresponding operators.
𝐻ˆ = 𝐻ˆ0 + 𝐻ˆ 0 (22.27) Landau, L. D. and Lifshitz, E. M. (1975).
The Classical Theory of Fields (4th edition).
where 𝐻ˆ 0 is the interaction Hamiltonian, which plays the role of the Pergamon Press, Oxford
𝑒 hˆ ® i 𝑒2 2
𝐻ˆ 0 = − 𝑝®𝐴 + (𝐴® 𝑝)
®ˆ + 𝐴 (22.28)
2𝑚𝑐 2𝑚𝑐 2
This expression can be somewhat simplified if we put, in accordance with
(20.27),
𝑝®ˆ 𝐴® (®
𝑟 ) − 𝐴® (®
𝑟 ) 𝑝® = −𝑖ℏ div 𝐴® (®
𝑟)
𝑒 ˆ ® 𝑒2 2
𝐻ˆ 0 = − (𝑝® 𝐴) + 𝐴 (22.29)
2𝑚𝑐 2𝑚𝑐 2
It should be noted here that the Hamiltonian (22.29) is responsible for all
processes of absorption and emission (spontaneous as well as induced) of
photons by an electron.
Obviously, the momentum operator in the momentum representation Momentum and Coordinate
is the momentum itself: Operators in the Momentum
Representation
𝑝ˆ𝑥 = 𝑝𝑥 𝑝®ˆ = 𝑝.
® (23.1)
The relation between the functions 𝜑 (𝑥) and Φ(𝑝𝑥 ), according to (15.6), is
of the form ∫
𝜑 (𝑥) = Φ (𝑝𝑥 ) 𝜓𝑝𝑥 (𝑥) 𝑑𝑝𝑥 (23.3)
where 𝜓𝑝𝑥 (𝑥) are the eigenfunctions of the operator 𝑝ˆ𝑥 in the coordinate
representation. By using (20.9) we can rewrite the expression (23.3) in the
following form:
∫
1
𝜑 (𝑥) = √ Φ (𝑝𝑥 ) exp(𝑖𝑝𝑥 𝑥/ℏ) 𝑑𝑝𝑥 (23.4)
2𝜋ℏ
𝑑
𝑥 Φ(𝑝𝑥 ) exp(𝑖𝑝𝑥 𝑥/ℏ) = −𝑖ℏ [Φ exp(𝑖𝑝𝑥 𝑥/ℏ)]
𝑑𝑝𝑥
(23.6)
𝑑Φ
+ 𝑖ℏ [Φ exp(𝑖𝑝𝑥 𝑥/ℏ)]
𝑑𝑝𝑥
We substitute (23.6) into (23.5) and consider the integral with respect to 𝑝𝑥 .
In so doing, we take into account that
∫∞
𝑑
−𝑖ℏ [Φ exp(𝑖𝑝𝑥 𝑥/ℏ)] 𝑑𝑝𝑥 = −𝑖ℏ Φ(𝑝𝑥 ) exp(𝑖𝑝𝑥 𝑥/ℏ)| ∞
−∞ = 0
𝑑𝑝𝑥
−∞
according to (15.17),
∫
1
exp 𝑖 (𝑝𝑥 − 𝑝𝑥0 )𝑥/ℏ 𝑑𝑥 = 𝛿 (𝑝𝑥 − 𝑝𝑥0 )
2𝜋ℏ
As a result, we are left with only the integral with respect to 𝑝𝑥 and (23.7)
assumes the form
∫ ∫
𝑑
𝜑 ∗ (𝑥) 𝑥 𝜑 (𝑥) 𝑑𝑥 = Φ∗ (𝑝𝑥 ) 𝑖ℏ Φ(𝑝𝑥 )𝑑𝑝𝑥 (23.8)
𝑑𝑝𝑥
Comparing the right-hand sides of (23.8) and (23.2), we find the expres-
sion for the 𝑥-coordinate operator in the momentum representation:
𝑑
𝑥ˆ (𝑝𝑥 ) = 𝑖ℏ (23.9)
𝑑𝑝𝑥
𝑟®ˆ (𝑝)
® = 𝑖ℏ∇𝑝® (23.10)
By using(23.1) and (23.9) it is easy to see that the commutators of Unitary Invariance of the
the operators of the coordinate and momentum components will be ex- Commutation Relations
actly the same in the momentum representation, as in the coordinate
representation [we are speaking of the expressions (20.24)-(20.26)]. This
conclusion may also be extended to the expressions (20.27)-(20.30). In
other words, the commutation relations are independent of the choice of
a representation, i.e. are unitary invariants. This is quite natural, if we re-
call that the mathematical fact of commutation of operators has a definite
physical meaning which, obviously, cannot change while going over from
one representation to another.
Going over to the momentum representation, we can write (20.11) in Schrödinger Equation in the
the form Momentum Representation
𝐻ˆ (𝑝𝑥 ) 𝜏𝐸 (𝑝𝑥 ) = 𝐸 𝜏𝐸 (𝑝𝑥 ) (23.11)
𝑝𝑥2
𝐸= (23.16)
2𝑚
The result (23.16) has been already mentioned in Section 1. It means that
a freely moving particle simultaneously possesses a definite energy and a
definite momentum; moreover, these quantities are related to each other
by the classical relation (23.16). In the case of a freely moving micro-
particle, the stationary state is also an eigenfunction of the momentum
operator. We emphasize that this can in no way be extended to bound
micro-particles (see the following example).
where 𝜓𝑥0 (𝑝𝑥 ) are the eigenfunctions of the operator 𝑥ˆ in the momentum
representation. By using the fact that 𝜓𝑥0 (𝑝𝑥 ) = 𝜓𝑝∗𝑥 (𝑥), and (20.9), we can
rewrite (23.17) in the following form:
∫𝑎
1
𝜏𝑛 (𝑝𝑥 ) = √ 𝜑𝑛 (𝑥) exp(−𝑖𝑝𝑥 𝑥/ℏ) 𝑑𝑥 (23.18)
2𝜋ℏ
0
𝑝 𝑎
cos2
𝑥
, 𝑛 is odd
4𝜋𝑎𝑛 2
2ℏ
|𝜏𝑛 (𝑝𝑥 )| 2 = (23.19)
𝑝 2 𝑎2
2
2 𝑝𝑥 𝑎
sin , 𝑛 is even
ℏ 𝜋 2𝑛 2 − 𝑥 2 2ℏ
ℏ
Thus, it has been rigorously shown that the stationary states (energy
levels) of a particle in the potential well are not characterized by a definite
momentum but by a corresponding definite de Broglie wavelength. We
remind the reader that this circumstance was qualitatively discussed in
Section 5 when we solved futility of a graphical representation of a bound
microparticle in the form of a classical wave in a resonator.
Summing up, we can compile a “scheme” for the transition from one A Scheme for the Transition
representation to another as follows: from the Coordinate to the
Momentum Representation
Let us consider a one-dimensional periodic potential 𝑈 (𝑥) satisfying The Band Structure of the
the condition Energy Spectrum, Brillouin
𝑈 (𝑥 + 𝑎) = 𝑈 (𝑥) (24.1) Zones
Following the second method in the scheme given at the end of the pre-
ceding section, we change over to the momentum representation. This
means that the potential 𝑈 (𝑥) should be expressed as an operator in the
momentum representation, 𝑈ˆ (𝑝𝑥 ). (In order to simplify the notation we
shall write 𝑝 for 𝑝𝑥 here.)
Õ
∞
𝑖2𝜋𝑛𝑥
𝑈 (𝑥) = 𝑈𝑛 exp −
𝑛=−∞
𝑎
Õ
∞
2𝜋𝑛ℏ 𝑑
𝑈ˆ (𝑝) = 𝑈𝑛 exp (24.2)
𝑛=−∞
𝑎 𝑑𝑝
𝑑
We shall now show that the operator exp 𝑝 1 is a displacement opera-
𝑑𝑝
tor with a finite displacement in 𝑝-space by the amount 𝑝 = 𝑝 1 . This is so,
as
𝑑𝜏 1 𝑑 2𝜏
𝜏 (𝑝 + 𝑝 1 ) = 𝜏 (𝑝) + 𝑝 1 (𝑝) + 𝑝 12 2 + . . .
𝑑𝑝 2! 𝑑𝑝
𝑑 1 2 𝑑2
= 1 + 𝑝1 + 𝑝 1 2 + . . . 𝜏 (𝑝)
𝑑𝑝 2! 𝑑𝑝
𝑑𝜏
= exp 𝑝 1
𝑑𝑝
Thus
𝑑
exp 𝑝 1 𝜏 = 𝜏 (𝑝 + 𝑝 1 ) (24.3)
𝑑𝑝
From (24.2) and (24.3) it follows that
Õ
∞
2𝜋ℏ
𝑈ˆ (𝑝) 𝜏 (𝑝) = 𝑈𝑛 𝜏 𝑝 + 𝑛 (24.4)
𝑛=−∞
𝑎
2𝜋ℏ 2𝜋ℏ
functions 𝜏 (𝑝), 𝜏 𝑝 − ,𝜏 𝑝 + , etc. Generally speaking, this
𝑎 𝑎
system consists of an infinite number of equations:
2
𝑝 + 2𝜋ℏ Õ
𝑎 2𝜋ℏ(𝑛 + 1)
2𝜋ℏ
− 𝐸 𝜏 𝑝 + + 𝑈𝑛 𝜏 𝑝 + =0
2𝑚 𝑎 𝑎
𝑛
2 Õ∞
𝑝 2𝜋ℏ𝑛
− 𝐸 𝜏 (𝑝) + 𝑈𝑛 𝜏 𝑝 + =0 (24.6)
2𝑚 𝑎
𝑛=−∞
2
𝑝 − 2𝜋ℏ Õ
𝑎 2𝜋ℏ(𝑛 − 1)
2𝜋ℏ
− 𝐸 𝜏 𝑝 − + 𝑈𝑛 𝜏 𝑝 + =0
2𝑚 𝑎 𝑎
𝑛
𝐸𝑚𝑖𝑛
𝑗 ≤ 𝐸 𝑗 (𝑝) ≤ 𝐸𝑚𝑎𝑥
𝑗 (24.9)
The inequalities (24.9) include the energy values for the microparticle
which constitute the 𝑗th energy band. If 𝐸𝑚𝑎𝑥 𝑚𝑖𝑛
𝑗−1 < 𝐸 𝑗 , we get a region of
unattainable energy values between the ( 𝑗 − 1)th and 𝑗th energy bands.
This region is usually called the forbidden band.
Since the above replacement does not change anything, we can say that
the momentum 𝑝 has physically different values within the limits of the
band:
ℏ ℏ
−𝜋 ≤𝑝 ≤𝜋 (24.11)
𝑎 𝑎
2𝜋ℏ
In other words, the 𝑝-space is split into intervals of length and one
𝑎
has to consider 𝑝 only within the limits of one individual interval. These
intervals are called the Brillouin zones. In this case we are dealing with
one-dimensional Brillouin zones. In general, the Brillouin zones are three-
dimensional; they often have a very complex configuration, which reflects
the specific nature of the periodic field under consideration.
energy bands and Brillouin zones form the basis of the modern electronic Theory of Solids (2nd edition). Cambridge
University Press, London
theory of solids (see, for example, Ziman13 and Jones14 ). 14 Jones, H. (1960). The Theory of
Brillouin Zones and Electronic States in
Crystals. North-Holland, Amsterdam
Bloch Functions. Let us consider the 𝑗th energy band. Figure 24.1 shows the dependence 𝐸 𝑗 (𝑝)
for this band. We choose some value 𝐸 0 from this band and denote by 𝑝 0 the corresponding value
Figure 24.1 of the momentum for motion to the right. The wave function of the chosen stationary
state is denoted through 𝜏 𝑗0 (𝑝). It can be easily seen that this function differs from zero only for 𝑝 =
𝑝 0 + 2𝜋ℏ𝑛/𝑎 [it can be seen from the figure that only at these points does the curve 𝐸 𝑗 (𝑝) intersect the
straight line 𝐸 = 𝐸 0 ]. Hence the function 𝜏 𝑗0 (𝑝) may be written in the form
Õ
2𝜋ℏ𝑛
𝜏 𝑗0 (𝑝) = Ω (𝑝) 𝛿 𝑝 + 0
−𝑝 (24.12)
𝑛
𝑎
Next, we go over to the coordinate representation by using the familiar rule (23.4):
∫+∞
1
𝜑 0𝑗 (𝑥) = √ 𝜏 𝑗0 (𝑝) exp(𝑖𝑝𝑥/ℏ) 𝑑𝑝
2𝜋ℏ
−∞
The explicit form of the function 𝑢 0𝑗 (𝑥) is not known [ in order to know it, we should have known the
explicit form of 𝑈 (𝑥)]. However, it can be seen from (24.13) with the period of the that the function
𝑢 0𝑗 (𝑥) is periodic field:
𝑢 0𝑗 (𝑥 + 𝑎) = 𝑢 0𝑗 (𝑥) (24.14)
Thus the wave function of a stationary state given by the indices 𝑗 and 𝑝 has the following form in
coordinate representation [cf. (22.24)]
𝑖𝑝𝑥
𝜑 𝑗𝑝 (𝑥) = 𝑢 𝑗𝑝 (𝑥) exp (24.15)
ℏ
This is a plane wave [exp(𝑖𝑝𝑥/ℏ)] whose amplitude [𝑢 𝑗𝑝 (𝑥)] is periodic with the period of the field.
The functions (24.15) are referred to as Bloch functions in the literature.
Let us consider the motion of a particle is a field whose potential is The Kronig-Penney Potential
shown in Figure 24.2 (the so-called Kronig-Penney potential). This is the
simplest case of a periodic potential.
Figure 24.2 shows three spatial regions. Assuming first Figure 24.2 that
𝐸 > 𝑈 0 we write the solution equation (20.13):
for region 1
Figure 24.2: Periodic Kronig-Penney
𝜑 1 (𝑥) = 𝐴1 exp(𝑖𝑘 1𝑥) + 𝐵 1 exp(−𝑖𝑘 1𝑥) potential.
p
2𝑚(𝐸 − 𝑈 0 )
𝑘1 =
ℏ
for region 2
or, finally,
𝑖𝑝𝑙
𝜑 3 (𝑥) = exp {𝐴1 exp [𝑖𝑘 1 (𝑥 − 𝑙)] + 𝐵 1 exp [−𝑖𝑘 1 (𝑥 − 𝑙)]} (24.18)
ℏ
By using (24.18) and the expression for 𝜑 1 and 𝜑 2 , we can write the conti-
nuity conditions for the wave function and its first derivative at the points
corresponding to the potential jump (the points 𝑥 = 0 and 𝑥 = 𝑎). These
conditions form a homogeneous system of linear equations in terms of the
coefficients 𝐴1, 𝐵 1, 𝐴2, 𝐵 2 :
𝐴1 + 𝐵 1 = 𝐴2 + 𝐵 2
𝐴1 exp (𝑖𝑘 2𝑎) + 𝐵 1 exp (−𝑖𝑘 2𝑎) = exp (𝑖𝑝𝑙/ℏ) [𝐴1 exp (−𝑖𝑘 1𝑏) + 𝐵 1 exp (𝑖𝑘 1𝑏)],
𝑘 1 𝐴1 − 𝑘 1 𝐵 1 = 𝑘 2 𝐴2 − 𝑘 2 𝐵 2 ,
𝑘 2 [𝐴2 exp(𝑖𝑘 2𝑎) − 𝐵 2 exp(−𝑖𝑘 2𝑎)] = exp(𝑖𝑝𝑙/ℏ) [𝐴1 exp(−𝑖𝑘 1𝑏) − 𝐵 1 exp(𝑖𝑘 1𝑏)]𝑘 1
Since the modulus of the cosine cannot be greater than unity, we get the
following condition imposed on quantities 𝑘 1 and 𝑘 2 and, hence, on 𝐸:
2
𝑘 + 𝑘 22
−1 ⩽ cos(𝑘 2𝑎) cos(𝑘 1𝑏) − 1 sin(𝑘 2𝑎) sin(𝑘 1𝑏) ⩽ 1 (24.20)
2𝑘 1𝑘 2
This condition defines the allowed energy bands. We next consider the
case when 𝐸 < 𝑈 0 .
𝑏 𝑎 (𝑎 ≈ 𝑙); 𝐸 𝑈0 (24.23)
(the barriers are narrow and high). Since in this case the quantity 𝑏 can
become arbitrarily small, we can require that the following conditions be
fulfilled:
p
𝑏 2𝑚𝑈 0 /ℏ 1, or 𝑘 3𝑏 1 (24.24)
By taking (24.24) into account, we put cosh(𝑘 3𝑏) ≊ 1 and sinh(𝑘 3𝑏) ≊ 𝑘 3𝑏,
and besides, according to (24.23),
r
𝑘 32 − 𝑘 22 𝑘3 1 𝑈0
≈ ≈
2𝑘 2𝑘 3 2𝑘 2 2 𝐸
−1 ⩽ 𝐹 (𝑘 2𝑎) ⩽ 1 (24.27)
The function 𝐹 (𝑘 2𝑎) is shown in Figure 24.3. The parts of the 𝑘 2𝑎 axis,
for which the condition (24.27) is satisfied have been shaded in the di-
agram. They correspond to the allowed energy bands (remember that
√
𝑘 2 = 2𝑚𝐸/ℏ).
The conversion of energy levels of an electron in the atom into energy Formation of Energy Bands
bands of an electron “collectivized” by the crystal may be seen as an effect as an Effect of the Removal of
of the removal of commutation degeneracy. Commutation Degeneracy
Δ𝐸
Δ𝜖 = (2𝑙 + 1)
𝑁
ℏ
For the system of sublevels to be discrete, it is essential Δ𝜖 > , where
𝜏
𝜏 is the lifetime of an electron in a crystal. In other words, the distance
between sublevels must be greater than the uncertainty in the energy of
the sublevel described by the relation (3.2). This means that the condition
ℏ(2𝑙 + 1)𝑁
<𝜏 (24.28)
Δ𝐸
We assume that a microparticle undergoes a transition from one Quantum Transitions and the
stationary state to another under the action of some external factor. How Principle of Superposition of
to find the probability of such a transition? The initial and the final states States
of the microparticle are described by functions of the type (20.16). For
example, let the initial state be given by
𝑖𝐸𝑛 𝑡
Ψ(𝑥, 𝑡) = 𝜑𝑛 (𝑥) exp −
ℏ
(we shall call it the “unperturbed” equation). The physical nature of ex-
ternal factor, which causes the quantum transition of the microparticle, is
arbitrary. In particular, it may be the interaction of the microparticle with
electromagnetic radiation. In the quantum theory apparatus such a factor
appears in the form of an interaction potential which must be added to
the “unperturbed” Hamiltonian 𝐻ˆ . In Section 22 in the example of the in-
teraction of an electron with radiation, this “addition” to the Hamiltonian
was interpreted as some perturbation and was denoted by 𝐻ˆ 0. We shall
use the same notation here. By taking into account the perturbation 𝐻ˆ 0,
we can rewrite the Schrödinger equation in the form
𝜕
𝑖ℏ Φ𝑛 = (𝐻ˆ + 𝐻ˆ 0) Φ𝑛 (25.2)
𝜕𝑡
This equation is called the “perturbed” equation. Its solutions Φ𝑛 are no
longer stationary states. Hence the index 𝑛 here does not fix the energy
level, but merely indicates the past history: the given “perturbed” state
has “emerged” from 𝑛th “unperturbed” state.
The reader is in fact familiar with all this. The remarks made above are
in agreement with those made in Section 10, regarding the relation (10.3),
which is essentially equivalent to the, relation (25.3).
Thus, the transition probability is, as expected, the square of the modu-
lus of the corresponding transition amplitude:
where
𝐸𝑚 − 𝐸𝑘
𝜔𝑚𝑘 = (25.8)
ℏ
Taking into account (25.7) we may rewrite (25.6) in the following final
form:
𝑑 𝑖 Õ
𝜒𝑛𝑚 = − 𝜒𝑛𝑘 𝑚 𝐻ˆ 0 𝑘 exp (𝑖𝜔𝑚𝑘 𝑡) (25.9)
𝑑𝑡 ℏ
𝑘
The perturbation is usually very small, which enables us to obtain Application of the Method of
an approximate solution of the system (25.9) by using the method of Perturbations to Computation
perturbations. A small perturbation means that the function Φ𝑛 may be of the Transition Probabilities
represented in the form
Φ𝑛 = Ψ𝑛 + Λ𝑛 (25.10)
According to (25.11), the small addition Λ is in turn split into additions dif-
(1)
fering in the order of smallness: the functions 𝜒𝑛𝑘 are of the same order
(2)
as the perturbation, 𝜒𝑛𝑘 are of the order of the square of perturbation, etc.
Substituting (25.11) into (25.9), we get
𝑑 (1) 𝑖
𝜒𝑛𝑚 = − 𝑚 𝐻ˆ 0 𝑛 exp(𝑖𝜔𝑛𝑚 𝑡) (25.13)
𝑑𝑡 ℏ
This is the approximate expression for the amplitudes 𝜒𝑛𝑚 , obtained in
the first-order approximation in the method of perturbations.
If it turns out that 𝑚 𝐻ˆ 0 𝑛 = 0, we must use the approximate
expression for the amplitudes in the second-order approximation in the
method of perturbations. It is obtained from (25.12) by retaining terms of
the second order in the perturbation:
𝑑 (2) 𝑖 Õ (1)
ˆ 0
𝜒𝑛𝑚 = − 𝜒𝑛𝑘 𝑚 𝐻 𝑛 exp(𝑖𝜔𝑚𝑘 𝑡) (25.14)
𝑑𝑡 ℏ
𝑘
∫ 2
(1) 2 1 𝑡
ˆ 0
(1)
𝑤𝑛𝑚 = 𝜒𝑛𝑚 (𝑡) = 2 𝑚 𝐻 (𝑡) 𝑛 exp(𝑖𝜔𝑚𝑛 𝑡) 𝑑𝑡
(25.16)
ℏ −∞
etc.
The further course of action envisages a substitution of definite opera- 15 Fermi, E. (1961). Notes on quantum me-
tors 𝐻ˆ 0 in expressions of the type (25.16) and (25.17). In quantum electron- chanics. University of Chicago Press
ics, for example, the operator (22.29) is used. A detailed consideration of 16 Tarasov, L. V. (1976). Physical Founda-
such questions is beyond the scope of this book. The reader is advised to tions of Quantum Electronics (in Russian).
Sov. Radio, Moscow
refer, for example, to Fermi15 and Tarasov16 , in this connection.
𝑈ˆ (𝑡, 𝑡 0 ) = 1 (26.2)
𝑈ˆ 𝑈ˆ† = 𝑈ˆ†𝑈ˆ = 1 (26.3)
where the operator 𝐻ˆ does not depend on time. By using (26.4) and (26.2),
we can easily find the form of the operator 𝑈ˆ (𝑡, 𝑡 0 ):
𝑖
𝑈ˆ (𝑡, 𝑡 0 ) = exp − (𝑡 − 𝑡 0 ) 𝐻ˆ (𝑥) (26.5)
ℏ
Let 𝐿(𝑥)
ˆ be the operator of some physical quantity in Schrödinger’s repre-
sentation. According to (17.32) and (26.7), this operator in the Heisenberg
representation will have the form
Suppose that the Hamiltonian of a microsystem can be broken into Interaction Representation
two components, one of which (𝐻ˆ0 ) represents the Hamiltonian of the (Dirac Representation)
microsystem itself, and the other (𝐻ˆ1 ) describes the interaction of the
initial microsystem with external fields or other systems (in other words,
is “responsible” for the effect of perturbation of the initial microsystem) :
In accordance with (17.32) we find from here the form of the operator
𝐿ˆ𝐵 (𝑥, 𝑡) in the interaction representation:
𝑖 𝑖
𝐿𝐵 (𝑥, 𝑡) = exp (𝑡 − 𝑡 0 )𝐻 0 𝐿 (𝑥) × exp − (𝑡 − 𝑡 0 )𝐻 0
ˆ ˆ ˆ ˆ (26.16)
ℏ ℏ
We emphasize that in (26.15) and (26.16) we have used not the entire
Hamiltonian, but just its “unperturbed” component 𝐻ˆ 0 (𝑥). Differentiating
(26.15) with respect to time, we get
𝜕 𝑖 𝜕
𝑖ℏ Ψ(𝑥, 𝑡) = −𝐻ˆ 0 (𝑥) Ψ(𝑥, 𝑡) + 𝑖ℏ exp (𝑡 − 𝑡 0 )𝐻ˆ0 (𝑥) 𝜓 (𝑥, 𝑡)
𝜕𝑡 ℏ 𝜕𝑡
Since
𝜕
𝑖ℏ 𝜓 (𝑥, 𝑡) = 𝐻ˆ 0 (𝑥) + 𝐻ˆ 1 (𝑥, 𝑡) 𝜓 (𝑥, 𝑡)
𝜕𝑡
the last result can be written in the following form:
𝜕 𝑖
𝑖ℏ Ψ(𝑥, 𝑡) = exp (𝑡 − 𝑡 0 )𝐻 0 (𝑥) 𝐻ˆ 1 (𝑥, 𝑡) 𝜓 (𝑥, 𝑡)
ˆ
𝜕𝑡 ℏ
𝑖 𝑖
= exp (𝑡 − 𝑡 0 ) 𝐻 0 (𝑥) 𝐻 1 exp − (𝑡 − 𝑡 0 ) 𝐻 0 (𝑥) Ψ(𝑥, 𝑡)
ˆ ˆ ˆ
ℏ ℏ
= 𝐻ˆ 1𝐵 (𝑥, 𝑡) Ψ(𝑥, 𝑡)
(26.17)
The vector analogy enables us to compare quite clearly all the three On the Vector Analogy Again
representations considered above. We correlate the system of basic states
of the microparticle with the system of mutually orthogonal basic vectors
in some arbitrary space. We shall consider all operators in matrix form
defined by the system of basic vectors. The states of the microparticle are
described by vectors considered in the coordinate system defined by the
basic vectors. Thus we have a system of basic vectors and a set of vector
states to be considered relative to this system.
The methods of describing the evolution of microsystems with time One Additional Remark
considered above are based on the use of the quantum-mechanical equa-
tions of motion. They assume a continuous evolution in time of either
amplitudes of states or of certain Hermitian operators, or simultaneously
of the amplitudes of states and the operators both. However, there are
other qualitatively different processes. Thus, it is well known that a de-
struction of the superposition of states caused by the detector in the act
of measurement leads to an abrupt change in the amplitude of state. It is
obvious that this change in the amplitude does not follow any equation of
motion and obeys only probabilistic predictions.
232 basi c c o n c e p t s o f q ua n t u m m e c h a ni c s
Nineteenth century was an era of rapid growth in physics. It is enough to “The Crisis in Physics”
mention just a few areas: the achievements in electricity and magnetism
which led to Maxwells electromagnetic field theory and permitted the
inclusion of optics into the framework of electromagnetic phenomena;
the significant progress in the development of classical mechanics which
came close to perfection as the result of a number of brilliant mathemat-
ical works; the enunciation of many universal principles in physics, of
prime importance among them being the law of conservation and trans-
formation of energy. It is not astonishing that towards the end of the
19th century it was generally believed that the description of the laws of
nature was in a final stage. In this respect the famous remarks of Planck
are worth noting. After defending his Ph.D. thesis, Planck wrote to his
teacher and mentor Philip Jolly asking his advice as to whether he should
seek a career in theoretical physics.
“Young man”, replied Jolly, “Why do you want to ruin your life? The the-
oretical physics is practically finished, the differential equations have all
been solved. AII that is left now is to consider individual special cases in-
volving variations of initial boundary conditions. Is it worthwhile taking up
a job which does not hold any prospects for the future?”
Events which followed dispelled such illusions very soon. At the turn
of the 20th century a number of fundamental discoveries which could not
be contained within the framework of the existing theories in physics
were reported. The list of these discoveries was quite imposing: 𝑋 -rays,
the dependence of the mass of an electron on its velocity, the incompre-
hensible laws of the photoelectric effect, radioactivity, etc. It appeared
that nature had decided to “laugh” at the self-confidence of people who
thought they had uncovered all its secrets.
Analysing the causes that led to this crisis, Lenin wrote: It is mainly
because the physicists did not know dialectics that the new physics strayed
into idealism.
‘Matter disappears’ means that the limit within which we have hitherto
known matter disappears and that our knowledge is penetrating deeper.
Lenin pointed out that the period of crisis will culminate in a new leap
in the development of physics and that its further development will occur
on the lines of materialistic dialectics. He wrote:
Looking back, we can now say that this “travail” led also to the birth of
quantum mechanics. As Lenin envisaged, the overcoming of this “crisis
in physics” resulted in a deepening of our knowledge of matter, and de-
manded a decisive turn from metaphysical to dialectical ideas. This was
a b r i e f h i s to r i c a l s u r v e y 235
§ The first stage: end of 19th century-1912 (first experiments and first
attempts to explain them).
The foundations of quantum mechanics were laid by experiments First Experiments and First
conducted at the end of the 19th century and the beginning of the 20th Attempts to Explain Them
century in various branches of physics which at that time were not con- (end of 19th century-1912)
nected with one another, e.g. atomic spectroscopy, study of black body
radiation and the photoelectric effect, solid state physics, study of the
structure of atom. By the end of the 19th century a lot of experimental
material on the radiation spectra of atoms was accumulated. It turned
out that atomic spectra are ordered sets of discrete lines (series). In 1885,
Balmer discovered a series of lines of atomic hydrogen, later named after
him, that could be described by a simple formula. In 1889, Rydberg found
a series of lines for thallium and mercury. Extensive studies of the spectra
of different atoms were conducted during this period by Kaiser and Runge
who used photographic methods. In 1904, Lyman discovered a series of
hydrogen lines falling in the ultraviolet region of the spectrum and in
1909, Paschen found a hydrogen series in the infrared region of the spec-
trum. Remarkably, the Lyman and Paschen series could be described by
a formula which was very close to the one established earlier by Balmer.
Noticing the regularities among various series of an atom, Ritz in 1908 for-
mulated his famous combination principle (see Section 2). However, right
until 1913, this formula could not be explained, the nature of the spectral
lines remained unclear.
heat of boron, carbon and silicon. The fact that specific heats of solids de-
pend on temperature could not be explained within the framework of clas-
sical theory until the appearance in 1907 of Einstein’s paper Correlating
Planck’s Radiation Theory and the Specific Heat Theory. Applying Planck’s
idea of quantization of energy to atomic vibrations in crystals, Einstein
deduced a formula which, in agreement with experiment, described the
temperature dependence of the specific heat of solids. Einstein’s work Iaid
the foundation of the modern theory of the specific heat of solids.
Thus, from the end of the 19th century to the year 1913 a large num-
ber of experimental facts, which could not be explained on the basis of
existing theory, were accumulated: the discovery of ordered series in
atomic spectra, the discovery of the quantization of energy in black body
radiation, the photoelectric effect, and the specific heat of solids; also
the planetary model of the atom was created. However, until 1913, all
these discoveries were considered separately. It was Bohr’s genius that
understood the common character of these facts and created a fairly har-
monious quantum theory on the atom based on these facts.
Bohr’s famous paper On the Constitution of Atoms and Molecules ap- Bohr’s Quantum Theory (1913-
peared in 1913. It considered the theory of the planetary model of the 1922)
hydrogen atom based on the idea of quantization (the energy and the an-
gular momentum of an electron in an atom were quantized). Resolutely
departing from accepted concepts, Bohr’s theory ruled out a direct link
between the frequency of the radiation emitted by an atom and the fre-
quency of the rotation of the electron in the atom. Having acquainted
himself with Bohr’s theory, Einstein remarked:
238 basi c c o n c e p t s o f q ua n t u m m e c h a ni c s
Then the frequency of light does not depend at all on the frequency of the
electron . . . This is an enormous achievement!
In 1923, Compton discovered the effect, later named after him, of a de- The Growth of Quantum Me-
crease in the wavelength of X-rays upon scattering by matter. This effect chanics (1923-1927).
clearly indicated the existence of wave as well as corpuscular properties
of radiation. Light quanta were introduced into physics as elementary
particles once and for all under the name of photons.
that the wave formalism of Schrödinger’s theory was very well received,
since it enabled a solution of quantum-mechanical problem with the help
of established methods of mathematical physics. Planck’s opinion19 of 19 Planck, M. (1928). Uber die ahhand-
lungen zur wellenmecbanik von erwin
Schrödinger equation is worth noting. According to him, the fundamental schrodinger (in German). Deutsche
importance of this differential equation lies not only in the way it has Ltteraturzeitung, 5:59–62
been derived, but also in its physical interpretation, whose details are
still not clear. But most important is the fact that owing to the introduc-
tion of the quantum law into the well-known system of usual differential
equations, we get an entirely new method which, with the help of mathe-
matics, can solve the complicated quantum-mechanical problem. This is
the first case when a quantum of action, which thus far was impervious to
all attempts to look at it from the point of view of classical physics, can be
included in a differential equation.
that the wave motion takes place not in the ordinary three-dimensional
space, but in the so called configurational space, where the dimensionality
is determined by the number of degrees of freedom of the system being
considered.
or ∫ ∫
𝜓 (𝑥) 𝐿ˆ∗ 𝜓 ∗ (𝑥) 𝑑𝑥 = 𝜆 ∗ 𝜓 (𝑥) 𝜓 ∗ (𝑥) 𝑑𝑥
Thus
∫ ∫
𝜓 ∗ (𝑥) 𝐿ˆ − 𝐿ˆ† 𝜓 (𝑥) 𝑑𝑥 = (𝜆 − 𝜆 ∗ ) 𝜓 ∗ (𝑥) 𝜓 (𝑥) 𝑑𝑥
It can be seen from here that the equality 𝜆 − 𝜆 ∗ (denoting the real positive
eigenvalues) is satisfied if and only if the operator 𝐿ˆ is Hermitian (i.e. if
and only if 𝐿ˆ − 𝐿ˆ† ).
∗ ∗ ∗
𝐿ˆ 𝜓𝑛 = 𝜆𝑛 𝜓𝑛 , 𝐿ˆ∗ 𝜓𝑚 = 𝜆𝑚 𝜓𝑚
or
∫ ∫
∗ ∗
𝜓𝑚 (𝑥) 𝐿ˆ 𝜓𝑛 (𝑥)𝑑𝑥 = 𝜆𝑛 𝜓𝑚 (𝑥) 𝜓𝑛 (𝑥)𝑑𝑥
∫ ∫
∗ ∗ ∗
𝜓𝑛 (𝑥) 𝐿ˆ∗ 𝜓𝑚 (𝑥)𝑑𝑥 = 𝜆𝑚 𝜓𝑛 (𝑥) 𝜓𝑚 (𝑥)𝑑𝑥
∗
If the operator 𝐿ˆ is Hermitian, then 𝐿ˆ = 𝐿ˆ† (and also (𝜆𝑚 = 𝜆𝑚 ). In this
case the last equality acquires the form
∫
∗
(𝜆𝑛 − 𝜆𝑚 ) 𝜓𝑚 𝜓𝑛 𝑑𝑥 = 0
Since 𝜆𝑛 ≠ 𝜆𝑚 , we get ∫
∗
𝜓𝑚 𝜓𝑛 𝑑𝑥 = 0
ap p e n d i c e s 245
The vector 𝑘® is directed along the light beam. At every point in space the
light beam is perpendicular to the surface of constant phase (i.e. perpen-
dicular to the surface Φ = const).
𝜕𝑆
𝑝® = ∇𝑆, 𝐸=− (B.3)
𝜕𝑡
The trajectories are lines perpendicular to the surface of constant action
(in the same way as the light rays are perpendicular to the surface of
constant phase). Comparing (B.2) and (B.3) and taking into account that
® we find that
𝑝® = ℏ𝑘,
𝑆
Φ=
ℏ
Thus the quasi-classical wave function is of the form
𝑖𝑆
𝜓 = 𝑎 exp (B.4)
ℏ
𝑖𝑆
get 𝑝® = ∇𝑆, 𝜓 = 𝑎 exp Thus the above equation assumes the form
ℏ
𝑖𝑆 𝑖𝑆
𝛾 ∇ exp = ∇𝑆 exp
ℏ ℏ
or
𝑖𝑆 𝑖 𝑖𝑆
𝛾 exp ∇𝑆 = ∇𝑆 · exp
ℏ ℏ ℏ
It can be easily seen that 𝑖 𝛾/ℏ = 1, and, consequently, 𝛾 = −𝑖ℏ. The quasi-
classical case may he used to substantiate the Schrödinger equation. It can
be easily seen that the well-known classical Hamilton-Jacobi equation
𝜕𝑆 1
+ (∇𝑆) 2 + 𝑈 = 0
𝜕𝑡 2𝑚
By cancelling out the factor exp(𝑖𝑆/ℏ), this equation becomes the Hamilton-
Jacobi equation.
C Commutation Relations
where 𝑒𝑖𝑘𝑛 is a unit antisymmetric tensor of the 3rd rank (𝑒 123 = 𝑒 231 =
𝑒 312 = 1, 𝑒 132 = 𝑒 321 = 𝑒 213 = −1, all the ether components of the tensor are
equal to zero).
Let us consider the commutator 𝑀ˆ 𝑖 , 𝑟ˆ𝑗 . By using (C.1), we can write
ap p e n d i c e s 247
Thus
𝑀ˆ 1, 𝑝ˆ2 = 𝑖 ℏ 𝑝ˆ3
𝑀ˆ 2, 𝑝ˆ3 = 𝑖 ℏ 𝑝ˆ1
𝑀ˆ 3, 𝑝ˆ1 = 𝑖 ℏ 𝑝ˆ2
𝑀ˆ 1, 𝑝ˆ1 = 𝑀ˆ 2, 𝑝ˆ2 = 𝑀ˆ 3, 𝑝ˆ3 = 0
Going over to the commutator 𝑀ˆ 𝑖 , 𝑀ˆ 𝑗 , we shall restrict ourselves for
248 basi c c o n c e p t s o f q ua n t u m m e c h a ni c s
we can write
𝑀ˆ 1, 𝑀ˆ 2 = (𝑟ˆ2 𝑝ˆ3 − 𝑟ˆ3 𝑝ˆ2 ) (𝑟ˆ3 𝑝ˆ1 − 𝑟ˆ1 𝑝ˆ3 ) − (𝑟ˆ3 𝑝ˆ1 − 𝑟ˆ1 𝑝ˆ3 ) × (𝑟ˆ2 𝑝ˆ3 − 𝑟ˆ3 𝑝ˆ2 )
= 𝑟ˆ2 𝑝ˆ3 𝑟ˆ3 𝑝ˆ1 − 𝑟ˆ2 𝑝ˆ3 𝑟ˆ1 𝑝ˆ3 − 𝑟ˆ3 𝑝ˆ2, 𝑟ˆ3 𝑝ˆ1 + 𝑟ˆ3 𝑝ˆ2 𝑟ˆ1 𝑝ˆ3 − 𝑟ˆ3 𝑝ˆ1 𝑟ˆ2 𝑝ˆ3
+ 𝑟ˆ3 𝑝ˆ1 𝑟ˆ3 𝑝ˆ2 + 𝑟ˆ1 𝑝ˆ3 𝑟ˆ2 𝑝ˆ3 − 𝑟ˆ1 𝑝ˆ3 𝑟ˆ3 𝑝ˆ2
Note that 𝑟ˆ2 𝑝ˆ3 𝑟ˆ1 𝑝ˆ3 = 𝑟ˆ1 𝑝ˆ3 𝑟ˆ2 𝑝ˆ3 and 𝑟ˆ3 𝑝ˆ2 𝑟ˆ3 𝑝ˆ1 = 𝑟ˆ3 𝑝ˆ1 𝑟ˆ3 𝑝ˆ2 . By taking this
into account, we write the expression for 𝑀ˆ 1, 𝑀ˆ 2 :
𝑀ˆ 1, 𝑀ˆ 2 = (𝑟ˆ2 𝑝ˆ3 𝑟ˆ3 𝑝ˆ1 − 𝑟ˆ3 𝑝ˆ1 𝑟ˆ2 𝑝ˆ3 ) + (𝑟ˆ3 𝑝ˆ2 𝑟ˆ1 𝑝ˆ3 − 𝑟ˆ1 𝑝ˆ3 𝑟ˆ3 𝑝ˆ2 )
= 𝑟ˆ2 𝑝ˆ1 (𝑝ˆ3 𝑟ˆ3 − 𝑟ˆ3 𝑝ˆ3 ) + 𝑟ˆ1 𝑝ˆ2 (𝑟ˆ3 𝑝ˆ3 − 𝑝ˆ3 𝑟ˆ3 )
D Commutation of Operators 𝑀ˆ 2 , 𝑀ˆ 𝑖
[𝑀ˆ 2, 𝑀ˆ 1 ] = ( 𝑀ˆ 12 + 𝑀ˆ 22 + 𝑀ˆ 32 ) 𝑀ˆ 1 − ( 𝑀ˆ 1 𝑀ˆ 12 + 𝑀ˆ 22 + 𝑀ˆ 32 )
= ( 𝑀ˆ 22 + 𝑀ˆ 32 ) 𝑀ˆ 1 − 𝑀ˆ 1 (𝑀ˆ 22 + 𝑀ˆ 32 )
= 𝑀ˆ 2 𝑀ˆ 2 𝑀ˆ 1 + 𝑀ˆ 3 𝑀ˆ 3 𝑀ˆ 1 − 𝑀ˆ 1 𝑀ˆ 2 𝑀ˆ 2 − 𝑀ˆ 1 𝑀ˆ 3 𝑀ˆ 3
= (𝑀ˆ 2 𝑀ˆ 1 𝑀ˆ 2 − 𝑖 ℏ 𝑀ˆ 2 𝑀ˆ 3 ) + ( 𝑀ˆ 3 𝑀ˆ 1 𝑀ˆ 3 + 𝑖 ℏ 𝑀ˆ 3 𝑀ˆ 2 )
− ( 𝑀ˆ 2 𝑀ˆ 1 𝑀ˆ 2 + 𝑖ℏ 𝑀ˆ 3 𝑀ˆ 2 ) − ( 𝑀ˆ 3 𝑀ˆ 1 𝑀ˆ 3 − 𝑖 ℏ 𝑀ˆ 2 𝑀ˆ 3 )
=0
1 Õ
∞
√ = 𝑃𝑙 (𝑥) 𝑡 𝑛 (E.1)
1 + 𝑡 2 − 2𝑡𝑥 𝑙=0
and satisfy the condition 𝑦 (1) = 1. They can he written in the form
1 𝑑𝑙 h 2 𝑙
i
𝑃𝑙 (𝑥) = (𝑥 − 1) (E.3)
2𝑙 𝑙! 𝑑𝑥 𝑙
We note that
∫1
2
𝑃𝑙 (𝑥) 𝑃𝑙 0 (𝑥) 𝑑𝑥 = 𝛿𝑙𝑙 0 (E.4)
2𝑙 + 1
−1
𝑙
The functions 𝑝𝑚 (𝑥) satisfy the differential equation
𝑚2
(1 − 𝑥 2 )𝑦 00 − 2𝑥𝑦 0 + 𝑙 (𝑙 + 1) − 𝑦=0 (E.6)
1 − 𝑥2
∫1
2 (𝑙 + 𝑚)!
𝑝𝑙𝑚 (𝑥) (𝑥) 𝑝𝑙𝑚0 (𝑥) 𝑑𝑥 = 𝛿𝑙𝑙 0 = 0 (E.7)
2𝑙 + 1 (𝑙 − 𝑚)!
−1
Δ𝜃 𝜑 𝑦 + 𝑙 (𝑙 + 1)𝑦 = 0 (E.9)
where
1 𝜕 1 𝜕 𝜕
Δ𝜃 𝜑 = + sin 𝜃
sin2 𝜃 𝜕𝜑 2 sin 𝜃 𝜕𝜃 𝜕𝜃
The spherical functions are orthonormalized:
∫2𝜋 ∫𝜋
∗
𝑌𝑙𝑚 (𝜃, 𝜑) 𝑌𝑙 0𝑚0 (𝜃, 𝜑) sin 𝜃 𝑑𝜃 𝑑 𝜑 = 𝛿𝑙𝑙 0 𝛿𝑚𝑚0 (E.10)
0 0
Õ
∞
𝑡𝑛
exp (2𝑥𝑡 − 𝑡 2 ) = 𝐻𝑛 (𝑥) (E.11)
𝑛=0
𝑛!
𝑑𝑛
𝐻𝑛 (−𝑥) = (−1)𝑛 exp (𝑥 2 ) exp (−𝑥 2 ) (E.12)
𝑑𝑥 𝑛
Some of the values are
∫∞
√
𝐻𝑛 (𝑥) 𝐻𝑛0 (𝑥) exp −𝑥 2 𝑑𝑥 = 𝜋 2𝑛 𝑛! 𝛿𝑛𝑛0 (E.14)
−∞
2 𝑥 𝐻𝑛 − 2𝑛 𝐻𝑛−1 = 𝐻𝑛+1
2𝑛 𝐻𝑛−1 = 𝐻𝑛0
𝑦 00 + (2𝑛 + 1 − 𝑥 2 )𝑦 = 0 (E.17)
252 basi c c o n c e p t s o f q ua n t u m m e c h a ni c s
2𝑚 𝑚 2𝜔 2 2
𝜑 00 (𝑥) + 2
𝐸 𝜑 (𝑥) − 𝑥 𝜑 (𝑥) = 0 (F.1)
ℏ ℏ2
We introduce the notation
r
𝑚𝜔 2𝐸
𝜉 =𝑥 , 𝜆=
ℏ ℏ𝜔
The factor 𝐶 is determined from the normalization condition for the wave
function 𝜑𝑛 (𝑥):
∫∞
𝜑𝑛2 (𝑥) 𝑑𝑥 = 1
−∞
Thus
∫∞
1= 𝜑𝑛2 (𝑥) 𝑑𝑥
−∞
∫∞
𝑑𝑥
= 𝜑𝑛2 (𝜉) 𝑑𝜉
𝑑𝜉
−∞
r ∫∞
ℏ
= 𝜑𝑛2 (𝜉) 𝑑𝜉
𝑚𝜔
−∞
r ∫∞
ℏ 2
= 𝐶 ℎ𝑛2 (𝜉) 𝑑𝜉
𝑚𝜔
−∞
𝑚𝜔 ℏ
ap p e n d i c e s 253
Bohr, N. (1955). Science and the unity of knowledge. In Leary, L., editor,
The Unity of Knowledge, pages 47–62. Doubleday.
Bohr, N. (1958a). Atomic Physics and Human Knowledge, volume 21. New
York, Wiley.
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