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Title: Lectures on quantum mechanics / Berthold-Georg Englert.
Description: Second edition, corrected and enlarged. | Hackensack : World Scientific Publishing
Co. Pte. Ltd., 2024. | Includes bibliographical references and index. |
Contents: Basic matters -- Simple systems -- Perturbed evolution.
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ISBN 9789811284755 (v. 2 ; hardcover) | ISBN 9789811284991 (v. 2 ; paperback) |
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Printed in Singapore
To my teachers, colleagues, and students
This page intentionally left blank
 Preface

This book on the Basic Matters of quantum mechanics grew out of a set of
lecture notes for a second-year undergraduate course at the National Uni-
versity of Singapore (NUS). It is a first introduction that does not assume
any prior knowledge of the subject. The presentation is rather detailed and
does not skip intermediate steps that — as experience shows — are not so
obvious for the learning student.
Starting from the simplest quantum phenomenon, the Stern–Gerlach
experiment with its choice between two discrete outcomes, and ending with
the standard examples of one-dimensional continuous systems, the physical
concepts and notions as well as the mathematical formalism of quantum
mechanics are developed in successive small steps, with scores of exercises
along the way. The presentation is “modern,” a dangerous word, in the
sense that the natural language of the trade — Dirac’s kets and bras and
all that — is introduced early, and the temporal evolution is dealt with
in a picture-free manner, with Schrödinger’s and Heisenberg’s equations of
motion side by side and on equal footing.
Two companion books on Simple Systems and Perturbed Evolution
cover the material of the subsequent courses at NUS for third- and fourth-
year students, respectively. The three books are, however, not strictly se-
quential but rather independent of each other and largely self-contained.
In fact, there is quite some overlap and a considerable amount of repeated
material. While the repetitions send a useful message to the self-studying
reader about what is more important and what is less, one could do with-
out them and teach most of Basic Matters, Simple Systems, and Perturbed
Evolution in a coherent two-semester course on quantum mechanics.
All three books owe their existence to the outstanding teachers, col-
leagues, and students from whom I learned so much. I dedicate these lec-
tures to them.

vii
viii Lectures on Quantum Mechanics: Basic Matters

I am grateful for the encouragement of Professors Choo Hiap Oh and

Kok Khoo Phua who initiated this project. The professional help by the
staff of World Scientific Publishing Co. was crucial for the completion; I
acknowledge the invaluable support of Miss Ying Oi Chiew and Miss Lai
Fun Kwong with particular gratitude. But nothing would have come about,
were it not for the initiative and devotion of Miss Jia Li Goh who turned
the original handwritten notes into electronic files that I could then edit.
I wish to thank my dear wife Ola for her continuing understanding and
patience by which she is giving me the peace of mind that is the source of
all achievements.
Singapore, March 2006 BG Englert

Note on the second edition

The feedback received from students and colleagues, together with my own
critical take on the three companion books on quantum mechanics, sug-
gested rather strongly that the books would benefit from a revision. This
task has now been completed.
Many readers have contributed entries to the list of errata. I wish to
thank all contributors sincerely and extend special thanks to Miss Hong
Zhenxi and Professor Lim Hock.
In addition to correcting the errors, I tied up some loose ends and
brought the three books in line with the later volumes in the “Lectures
on . . . ” series. There is now a glossary, and the exercises, which were in-
terspersed throughout the text, are collected after the main chapters and
supplemented by hints.
The team led by Miss Nur Syarfeena Binte Mohd Fauzi at World Sci-
entific Publishing Co. contributed greatly to getting the three books into
shape. I thank them very much for their efforts.
Beijing and Singapore, November 2023 BG Englert
 Contents

Preface vii

Glossary xiii
Miscellanea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Latin alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
Greek alphabet and Greek-Latin combinations . . . . . . . . . . xv

1. A Brutal Fact of Life 1

1.1 Causality and determinism . . . . . . . . . . . . . . . . . . 1
1.2 Bell’s inequality: No hidden determinism . . . . . . . . . . 4
1.3 Remarks on terminology . . . . . . . . . . . . . . . . . . . 7

2. Kinematics: How Quantum Systems are Described 9

2.1 Stern–Gerlach experiment . . . . . . . . . . . . . . . . . . 9
2.2 Successive Stern–Gerlach measurements . . . . . . . . . . 12
2.3 Order matters . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Mathematization . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Probabilities and probability amplitudes . . . . . . . . . . 20
2.6 Quantum Zeno effect . . . . . . . . . . . . . . . . . . . . . 29
2.7 Kets and bras . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.8 Brackets, bra-kets, and ket-bras . . . . . . . . . . . . . . . 34
2.9 Pauli operators, Pauli matrices . . . . . . . . . . . . . . . 38
2.10 Functions of Pauli operators . . . . . . . . . . . . . . . . . 40
2.11 Eigenvalues, eigenkets, and eigenbras . . . . . . . . . . . . 42
2.12 Wave–particle duality . . . . . . . . . . . . . . . . . . . . 46
2.13 Expectation value . . . . . . . . . . . . . . . . . . . . . . . 47
2.14 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

ix
x Lectures on Quantum Mechanics: Basic Matters

2.15 Statistical operator, Born rule . . . . . . . . . . . . . . . . 51

2.16 Mixtures and blends . . . . . . . . . . . . . . . . . . . . . 55
2.17 Nonselective measurement . . . . . . . . . . . . . . . . . . 55
2.18 Entangled atom pairs . . . . . . . . . . . . . . . . . . . . . 57
2.19 State reduction, conditional probabilities . . . . . . . . . . 64
2.20 Measurement outcomes do not pre-exist . . . . . . . . . . 66
2.21 Measurements with more than two outcomes . . . . . . . 68
2.22 Unitary operators . . . . . . . . . . . . . . . . . . . . . . . 73
2.23 Hermitian operators . . . . . . . . . . . . . . . . . . . . . 76
2.24 Hilbert spaces for kets and bras . . . . . . . . . . . . . . . 77

3. Dynamics: How Quantum Systems Evolve 79

3.1 Schrödinger equation . . . . . . . . . . . . . . . . . . . . . 79
3.2 Heisenberg equation . . . . . . . . . . . . . . . . . . . . . 83
3.3 Equivalent Hamilton operators . . . . . . . . . . . . . . . 85
3.4 Von Neumann equation . . . . . . . . . . . . . . . . . . . 86
3.5 Example: Larmor precession . . . . . . . . . . . . . . . . . 87
3.6 Time-dependent probability amplitudes . . . . . . . . . . 90
3.7 Schrödinger equation for probability amplitudes . . . . . . 91
3.8 Time-independent Schrödinger equation . . . . . . . . . . 95
3.9 Example: Two magnetic silver atoms . . . . . . . . . . . . 97

4. Motion Along the x Axis 105

4.1 Kets, bras, and wave functions . . . . . . . . . . . . . . . 105
4.2 Position operator . . . . . . . . . . . . . . . . . . . . . . . 110
4.3 Momentum operator . . . . . . . . . . . . . . . . . . . . . 112
4.4 Heisenberg’s commutation relation . . . . . . . . . . . . . 114
4.5 Position–momentum transformation function . . . . . . . 115
4.6 Expectation values . . . . . . . . . . . . . . . . . . . . . . 117
4.7 Uncertainty relation . . . . . . . . . . . . . . . . . . . . . 120
4.8 State of minimum uncertainty . . . . . . . . . . . . . . . . 123
4.9 Time dependence . . . . . . . . . . . . . . . . . . . . . . . 126
4.10 Excursion into classical mechanics . . . . . . . . . . . . . 127
4.11 Hamilton operator, Schrödinger equation . . . . . . . . . . 131
4.12 Time transformation function . . . . . . . . . . . . . . . . 133
 Contents xi

5. Elementary Examples 135

5.1 Force-free motion . . . . . . . . . . . . . . . . . . . . . . . 135
5.1.1 Time-transformation functions . . . . . . . . . . . . 135
5.1.2 Spreading of the wave function . . . . . . . . . . . 137
5.1.3 Long-time and short-time behavior . . . . . . . . . 142
5.1.4 Interlude: General position-dependent force . . . . 148
5.1.5 Energy eigenstates . . . . . . . . . . . . . . . . . . 150
5.2 Constant force . . . . . . . . . . . . . . . . . . . . . . . . 153
5.2.1 Energy eigenstates . . . . . . . . . . . . . . . . . . 153
5.2.2 Limit of no force . . . . . . . . . . . . . . . . . . . 155
5.3 Harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . 158
5.3.1 Energy eigenstates: Power-series method . . . . . . 158
5.3.2 Energy eigenstates: Ladder-operator approach . . . 163
5.3.3 Hermite polynomials . . . . . . . . . . . . . . . . . 168
5.3.4 Infinite matrices . . . . . . . . . . . . . . . . . . . . 170
5.3.5 Position and momentum spreads . . . . . . . . . . 172
5.4 Delta potential . . . . . . . . . . . . . . . . . . . . . . . . 173
5.4.1 Bound state . . . . . . . . . . . . . . . . . . . . . . 173
5.4.2 Scattering states . . . . . . . . . . . . . . . . . . . 178
5.5 Square-well potential . . . . . . . . . . . . . . . . . . . . . 182
5.5.1 Bound states . . . . . . . . . . . . . . . . . . . . . . 182
5.5.2 Delta potential as a limit . . . . . . . . . . . . . . . 186
5.5.3 Scattering states and tunneling . . . . . . . . . . . 187
5.6 Stern–Gerlach experiment revisited . . . . . . . . . . . . . 191

Exercises with Hints 195

Exercises for Chapters 1–5 . . . . . . . . . . . . . . . . . . . . . 195
Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

Index 217
This page intentionally left blank
 Glossary

Here is a list of the symbols used in the text; the numbers in square brackets
indicate the pages of first occurrence or of other significance.

Miscellanea
0 null symbol: number 0, or null column, or null matrix,
or null ket, or null bra, or null operator, et cetera
1 unit symbol: number 1, or unit matrix, or identity
operator, et cetera
A= bB read “A represents B” or “A is represented by B”
Max{ } , Min{ } maximum, minimum of a set of real numbers
(, ) inner product [77]
a∗ , a complex conjugate of a, absolute value of a
Re(a) , Im(a) real, imaginary part of a: a = Re(a) + i Im(a)
a= a length of vector a
a · b, a × b scalar, vector product of vectors a and b
A† adjoint of A [25]
det(A), tr(A) determinant [43], trace of A [49]
| i, h |; |1i, ha| generic ket, bra; labeled ket, bra [33]
| . . . , ti, h. . . , t| ket, bra at time t [81]
h | i, | ih | bra-ket, ket-bra [35]
h. . . , t1 |. . . , t2 i time transformation function [134]
hAi mean value, expectation value of A [48]
[A, B] commutator of A and B [83]
↑z , ↓z spin-up, spin-down in the z direction [33]
x! factorial of x [112]
2 −1
f (x), f (x)
x 7→ f (x):

square, inverse
of the function
f 2 (x) = f f (x) , f f −1 (x) = x, f −1 f (x) = x

xiii
xiv Lectures on Quantum Mechanics: Basic Matters

f (x)2 , f (x)−1 square, reciprocal

2 of the function value:
f (x)2 = f (x) , f (x)−1 = 1/f (x)
dt, δt differential, variation of t
d ∂
, total, parametric time derivative
dt ∂t
∇ gradient vector differential operator
⊗ tensor product: |ai ⊗ |bi = |a, bi [59]

Latin alphabet
a width of the square-well potential [182]
a, b detector settings [4]
ajk , bjk matrix elements [15]
(n)
ak power series coefficients [160]
A, A† harmonic-oscillator ladder operators [163]
A, B matrices [15]
B magnetic field [10]
B0 (t), b(t) on-axis field strength, near-axis gradient [191]
C(a, b) Bell correlation [5]
cos, sin, . . . trigonometric functions
cosh, sinh, . . . hyperbolic functions
e; ex = exp(x) Euler’s number, e = 2.71828 . . . ; exponential function
e, n unit vectors
ex , ey , ez unit vectors for the x, y, z directions [32]
Emagn ; Ek magnetic energy [10]; kth eigenenergy [95]
F flipper [17]
F,F force [11,128]
Ga , Gb generators [128]
h = 2π~ Planck’s constant,
~ = 1.05457 × 10−34 J s = 0.658212 eV fs [82]
H, H Hamilton operator [81], matrix representing H [92]
Hn ( ) nth Hermite polynomial [162]
i imaginary unit, i2 = −1
L coherence length [206]; Lagrange function [127]
M
mass [128]
O t4 terms of oder t4 or smaller [144]
prob(e) probability for event e [27]
p; p momentum [115]; at the stationary phase [156]
P, P (t) momentum operator [113], with time dependence [126]
 Glossary xv

r distance between atoms [97]

q distance in oscillator units [159]
r, t reflected, transmitted amplitude [188]
R,T single-photon counters [3]
Ry ( ) rotation matrix for the y axis [20]
s Bloch vector [54]
sgn(x) sign of x
S+ , S− +selector, −selector [16]
t; t0 , t1 time [81]; time before, after [80]
Uab , Ut (τ ), . . . unitary operators [73]
V0 depth of the square-well potential [182]
V (x) potential energy [128]
wk weights in blending ρ [121]
Wab ; W (p) action [127]; action variable [155]
x, y, z; ẋ cartesian coordinates [9]; velocity [127]
X, Z position operator for the x, z direction [111]
X(t), Z(t) time-dependent position operators [126]

Greek alphabet and Greek-Latin combinations

α phase shift [178]
α, β, . . . ; αk probability amplitudes [16]; kth amplitude [96]
δjk Kronecker’s delta symbol [69]
δ(x − x0 ) Dirac’s delta function [106]
δ(x − x0 ; ) model for Dirac’s delta function [108]
δA, (δA)2 spread, variance of A [120]
∆p(t), ∆z(t) transferred momentum, associated displacement [193]
(t) complex spreading factor [140]
η(x) Heaviside’s step function [208]
θ, ϑ energy parameters (square well) [183]
ϑ rotation angle [29]
κ reciprocal length [173]
κ, k energy parameters (square well) [182]
λ hidden variable [5]; eigenvalue candidate [22]
λ de Broglie wavelength [117]
λ+ , λ− measurement results [47]
Aλ (a), Bλ (b) measurement results [6]
w(λ) hidden-variable weight density [5]
Λ measurement symbol [47]
xvi Lectures on Quantum Mechanics: Basic Matters

µB ; µ Bohr magneton [202]; magnetic dipole moment

vector [10]
π Archimedes’s constant, π = 3.14159 . . .
(1)
ρ, ρ(pair) , ρT , . . . statistical operators [52]
σx , σy , σz ; σ Pauli operators for the x, y, z direction [39]; Pauli
vector operator [40]
σ (1) , σ (2) Tom’s, Jerry’s Pauli vector operator [57]
τ small time increment [81], path parameter [128]
t(τ ), x(τ ) path variables [128]
τ torque [10]
φ rotation angle [20]; complex phase [28]
φk , φ†k kth eigencolumn, its adjoint [96]
ϕ complex phase [28]
ϕ, ϑ azimuth, polar angle [44]
Φ hermitian phase operator [76]
χn (q) wave function factor [160]
ψ(t) time-dependent column of probability amplitudes [94]
ψ(x), ψ(p) position [105], momentum wave function [118]
ω, Ω; ω, Ω angular frequencies; vectors [85,87]
ω; ωkl oscillator frequency [158]; transition frequency [97]
 Chapter 1

A Brutal Fact of Life

1.1 Causality and determinism

Before their first encounter with the quantum phenomena that govern the
realm of atomic physics and sub-atomic physics, students receive a train-
ing in classical physics, where Newton’s∗ mechanics of massive bodies and
Maxwell’s† electromagnetism — the physical theory of the electromagnetic
field and its relation to the electric charges — give convincingly accurate
accounts of the observed phenomena. Indeed, almost all experiences of
physical phenomena that we are conscious of without the help of refined
instruments fit perfectly into the conceptual and technical framework of
these classical theories. It is instructive to recall two characteristic features
that are equally possessed by Newton’s mechanics and Maxwell’s electro-
magnetism: Causality and Determinism.
Causality is inference in time: Once you know the state of affairs —
physicists prefer to speak more precisely of the “state of the system” —
you can predict the state of affairs at any later time, and often also retro-
dict the state of affairs at earlier times. Having determined the relative
positions of the sun, earth, and moon and their relative velocities, we can
calculate highly precisely when the next lunar eclipse will happen (extreme
precision on short time scales and satisfactory precision for long time scales
also require good knowledge of the positions and velocities of the other
planets and their satellites, but that is a side issue here) or when the last
one occurred. Quite similarly, present knowledge of the strength and di-
rection of the electric and magnetic fields together with knowledge about
the motion of the electric charges enable us to calculate reliably the elec-
tromagnetic field configuration in the future or the past.
∗ Isaac Newton (1643–1727) † James Clerk Maxwell (1831–1879)

1
2 A Brutal Fact of Life

Causality, as we shall see, is also a property of quantal evolution: Given

the state of the system now, we can infer the state of the system later (but,
typically, not earlier). Such as there are Newton’s equation of motion in me-
chanics and Maxwell’s set of equations for the electromagnetic field, there
are also equations of motion in quantum mechanics: Schrödinger’s∗ equa-
tion, which is more in the spirit of Maxwell’s equations, and Heisenberg’s†
equation, which is more in Newton’s tradition.
We say that the classical theories are deterministic because the state of
the system uniquely determines all phenomena. When the positions and
velocities of all objects are known in Newton’s mechanics, the results of all
possible measurements are predictable, there is no room for any uncertainty
in principle. Likewise, once the electromagnetic field is completely speci-
fied and the positions and velocities of all charges are known in Maxwell’s
electromagnetic theory, all possible electromagnetic phenomena are fully
predictable.
Let us look at a somewhat familiar situation that illustrates this point
and will enable us to establish the difference in situation that we encounter
in quantum physics. You have all seen reflections of yourself in the glass
of a shopping window while at the same time having a good view of the
merchandise for sale. This is a result of the property of the glass sheet that
it partly transmits light and partly reflects it. In a laboratory version, we
could have 50% probability each for transmission and reflection:
glass
..................................................................... ...........
............. ...... ... ...
..... ... ... ...
..... light source .........................................
... ...
... ...
..... .
............
........................................................................
..... ..
..
......................... ....
.................................................. 100%
..........................................................
... ...
... ...
... ...
. ........................... ..
........................
...............................
...
.................
. ... ...............................
...
..
..
..............
.................... ..... .....
. ........................
............................ 50%
.. . .. .....
. .
. .
........
..
............................................ ... ... ..............................................
.............................
........
........................ .... .... .........
reflected 50% ..... .....
... ... transmitted
..
...........
intensity intensity (1.1.1)
A light source emits pulses of light, which are split in two by such a half-
transparent mirror, half of the intensity being transmitted and the other
half reflected. Given the properties of pulses emitted by the source and the
material properties of the glass, we can predict completely how much of
the intensity is reflected, how much is transmitted, how the pulse shape is
changed, and so forth — all these are implications of Maxwell’s equations.
But, we know that there is a different class of phenomena that reveal a
certain graininess of light: the pulses consist of individual lumps of energy
— “light quanta” or “photons.” (We are a bit sloppy with the terminology
∗ Erwin Schrödinger (1887–1961) † Werner Heisenberg (1901–1976)
 Causality and determinism 3

here; at a more refined level, photons and light quanta are not the same,
but that is irrelevant presently.) We become aware of the photons if we dim
the light source by so much that there is only a single photon per pulse. We
also register the reflected and transmitted light by single-photon counters:

.............
.............................................................
......
glass
...... ... ...........
..
...
... dim ...
.. single ... ...
... ...
... ...
.................................................
...
... light source
....... ..
..... ........................
..
..
..
..
..
..
..
..
..
..
....... .
photon
. ...................................
... ...
... ...
... ...
............... ..... ... ...
........................................................ ........................
........................ ..
........................... ....
......................
..................... .........................................................
....................
.. ... ... ........................
......... ........................
...... . ...............................................
. .... ....
... ...
........................ .....
...................... .....
..... .................. .. .... ..
.......... ..... ..... .
.........
..
..... .....
single-photon ......... single-photon
counter “R” counter “T”
(1.1.2)

What will be the fate of the next photon to arrive? Since it cannot split
in two, either the photon is transmitted as a whole or it is reflected as a
whole so that eventually one of the counters will register the photon. A
single photon, so to say, makes one detector click: either we register a click
of detector T or of detector R, but not of both.
What is important here is that we cannot predict which detector will
click for the next photon, all we know is the history of the clicks of the
photons that have already arrived. Perhaps a sequence such as

RRTRTTTRTR (1.1.3)

was the case for the last ten photons. In a long sequence, reporting the
detector clicks of very many photons, there will be about the same number
of T clicks and R clicks because it remains true that half of the intensity
is reflected and half transmitted. On the single-photon level, this becomes
a probabilistic fact: Each photon has a 50% chance of being reflected and
an equal chance of being transmitted. And this is all we can say about the
future fate of a photon approaching the glass sheet.
So, when repeating the experiment with another set of ten photons,
we do not reproduce the above sequence of detector clicks, but rather get
another one, perhaps

R T T R R R R T R T. (1.1.4)

And a third set of ten would give yet another sequence, all 210 possible
sequences occurring with the same frequency if we repeat the experiment
very often.
Thus, although we know exactly all the properties of the incoming pho-
ton, we cannot predict which detector will click. We can only make statis-
4 A Brutal Fact of Life

tical predictions that answer questions such as the following: “How likely
are four Ts and six Rs in the next sequence of ten?”
What we face here, in a simple but typical situation, is the lack of
determinism of quantum phenomena. Complete knowledge of the state
of affairs does not enable us to predict the outcomes of all measurements
that could be performed on the system. In other words, the state does not
determine the phenomena. There is a fundamental element of chance: The
laws of nature that connect our knowledge about the state of the system
with the observed phenomena are probabilistic, not deterministic.

1.2 Bell’s inequality: No hidden determinism

Now, that raises the question about the origin of this probabilistic nature:
Does the lack of determinism result from a lack of knowledge? Or, put
differently, could we know more than we do and then have determinism re-
installed? The answer is No. Even if we know everything that can possibly
be known about the photon, we cannot predict its fate.
It is not simple to make this point for the example discussed above
with single photons incident on a half-transparent mirror. In fact, one can
construct contrived formalisms in which the photons are equipped with
internal clockworks of some sort that determine in a hidden fashion where
each photon will go. But in more complicated situations, even the most
ingenious deterministic mechanism cannot reproduce the observed facts in
all respects. The following argument is a variant of the one given by Bell∗
in the 1960s.
Consider the more general scenario in which a photon-pair source always
emits two photons, one going to the left and the other going to the right:

............. ...............
+1 .......... ................... ................................................................ .................................................................................... ................................................................ ... ..
......... .....
.
+1
.. ......... ..... ... ... ... ... ...
... .................
.........
....
Setting ...
.
.
...............................................................
...
.
. photon-pair ...
.
.
...............................................................
...
.
. Setting .
...
...
..
..
.

....... ... ... ... ... ............

. ......... ..
.... ............. .....
a .
...
.
...
source ..
..
..
.... b ... ..........
.. ......... ......
..... ....... .............................................................. .................................................................................... ............................................................. ...... ..
−1 ...... .
.......
..
..............
.. −1
(1.2.1)
Each photon is detected by one of two detectors eventually — with measure-
ment results +1 or −1 — and the devices allow for a number of parameter
settings. We denote by symbol a the collection of parameters on the left,
and by b those on the right. Details do not matter; we just need that
different settings are possible, that there is a choice between different mea-
∗ John Stewart Bell (1928–1990)
 Bell’s inequality: No hidden determinism 5

surements on both sides. The only restriction we insist upon is that there
are only two possible outcomes for each setting, the abstract generaliza-
tion of “transmission” and “reflection” in the single-photon plus glass sheet
example above.
For any given setting, the experimental data are of this kind:
photon pair no. 1 2 3 4 5 6 7 8 ...
on the left +1 +1 −1 −1 +1 −1 +1 +1 ...
(1.2.2)
on the right +1 −1 −1 +1 −1 −1 −1 +1 ...
product +1 −1 +1 −1 −1 +1 −1 +1 ...

The products in the last row distinguish the pairs with the same outcomes
on the left and the right (product = +1) from those with opposite outcomes
(product = −1). We use these products to define the Bell correlation C(a, b)
for the chosen setting specified by parameters a, b,
(number of +1 pairs) − (number of −1 pairs)
C(a, b) = . (1.2.3)
total number of observed pairs
Clearly, we have C(a, b) = +1 if +1 on one side is always matched with a
+1 on the other and −1 with −1, and we have C(a, b) = −1 if +1 is always
paired with −1 and −1 with +1. In all other cases, the value of C(a, b) is
between these extrema, so that

−1 ≤ C(a, b) ≤ +1 (1.2.4)

for any setting a, b.

Following Bell, let us now fantasize about a (hidden) mechanism that
determines the outcome on each side. We conceive each pair as being
characterizable by a set of parameters collectively called λ and that the
source realizes various λ with different relative frequencies. Thus, there
is a positive weight function w(λ), such that dλ w(λ) is the probability of
having a λ value within a dλ volume around λ. These probabilities must
be positive numbers that sum up to unity,
Z
w(λ) ≥ 0 , dλ w(λ) = 1 . (1.2.5)

We need not be more specific because further details are irrelevant to the
argument — which is, of course, the beauty of it.
We denote by Aλ (a) the measurement result on the left for setting a
when the hidden control parameter has value λ and by Bλ (b) the corre-
sponding measurement result on the right. Since all measurement results
6 A Brutal Fact of Life

are either +1 or −1, we have

Aλ (a) = ±1 , Bλ (b) = ±1 for all a, b, λ (1.2.6)
and also

Aλ (a)Bλ (b) = ±1 for all a, b, λ . (1.2.7)

This is then the product to be entered in the table (1.2.2) for the pair
that leaves the source with value λ and encounters the settings a and b.
Upon summing over all pairs, we get
Z
C(a, b) = dλ w(λ)Aλ (a)Bλ (b) (1.2.8)

for the Bell correlation, and all the rest follows from this expression.
Before proceeding, however, let us note that an important assumption
has entered: We take for granted that the measurement result on the left
does not depend on the setting of the apparatus on the right, and vice versa.
This is an expression of locality as we naturally accept it as a consequence
of Einstein’s∗ observation that spatially well-separated events cannot be
connected by any causal links if they are simultaneous in one reference
frame. Put differently, if the settings a and b are decided very late, just
before the measurements actually take place, any influence of the setting
on one side upon the outcome on the other side would be inconsistent with
Einsteinian causality. With this justification, there is no need to consider
the more general possibility of having Aλ (a, b) on the left and Bλ (a, b) on the
right. Such a b dependence of Aλ and an a dependence of Bλ are physically
unacceptable, but of course, it remains a mathematical possibility that
cannot be excluded on purely logical grounds.
All together, we now consider two settings on the left, a and a0 , and two
on the right, b and b0 ; then, there are four Bell correlations. We subtract
the fourth from the sum of the other three,
C(a, b) + C(a, b0 ) + C(a0 , b) − C(a0 , b0 )
Z h

i
= dλ w(λ) Aλ (a) Bλ (b) + Bλ (b0 ) +Aλ (a0 ) Bλ (b) − Bλ (b0 ) ,
| {z } | {z }
= 0 or ± 2 = 0 or ± 2
(1.2.9)
where, for each value of λ, one of the Bλ (b) ± Bλ (b0 ) terms
  equals 0 and the
other term is ±2 as follows from (1.2.6). Therefore, · · · in (1.2.9) is ±2
∗ Albert Einstein (1879–1955)
 Remarks on terminology 7

for every λ, and since the integral is the average value, we conclude that

C(a, b) + C(a, b0 ) + C(a0 , b) − C(a0 , b0 ) ≤ 2 . (1.2.10)

This is (a variant of) the so-called Bell inequality. Given the very simple
argument and the seemingly self-evident assumptions entering at various
stages, one should confidently expect that it is generally obeyed by the
correlations observed in any experiment of the kind depicted in (1.2.1).
Anything else would defy common sense, would it not? But the fact is that
rather strong violations are observed in real-life experiments in which √
the
left-hand side substantially exceeds 2, getting very close indeed to 2 2,
the maximal value allowed for Bell correlations in quantum mechanics (see
Section 2.18).
Since we cannot possibly give up our convictions about locality, and
thus about Einsteinian causality, the logical conclusion must be that there
just is no such hidden deterministic mechanism [characterized by w(λ) as
well as Aλ (a) and Bλ (b)]. We repeat:
There is no mechanism that decides the
(1.2.11)
outcome of a quantum measurement.
What is true for such correlated pairs of photons is, by inference, also true
for individual photons. There is no mechanism that decides whether the
photon is transmitted or reflected by the glass sheet; it is rather a truly
probabilistic phenomenon.
This is a brutal fact of life. In a very profound sense, quantum mechanics
is about learning to live with it.

1.3 Remarks on terminology

We noted the fundamental lack of determinism at the level of quantum

phenomena and the consequent inability to predict the outcome of all ex-
periments that could be performed. It may be worth emphasizing that this
lack of predictive power is of a very different kind than, say, the impossi-
bility of forecasting next year’s weather.
The latter is a manifestation of the chaotic features of the underlying
dynamics, frequently referred to as deterministic chaos. In this context,
“deterministic” means that the equations of motion are differential equa-
tions that have a unique solution for given initial values — the property
that we called “causal” above. There is a clash of terminology here if one
wishes to diagnose one.
8 A Brutal Fact of Life

The deterministic chaos comes about because the solutions of the equa-
tions of motion depend extremely sensitively on the initial values, which
in turn are never known with utter precision. This sensitivity is a generic
feature of nonlinear equations and not restricted to classical phenomena.
Heisenberg’s equations of motion of an interacting quantum system are just
as nonlinear as Newton’s equations for the corresponding classical system
if there is one.
In classical systems that exhibit deterministic chaos, our inability to
make reliable predictions concerns phenomena that are sufficiently far away
in the future — a weather forecast for the next three minutes is not such
a challenge. In the realm of quantum physics, however, the lack of deter-
minism is independent of the time elapsed since the initial conditions were
established. Even perfect knowledge of the state of affairs immediately be-
fore a measurement is taken does not enable us to predict the outcome; at
best, we can make a probabilistic prediction, a statistical prediction.
Of course, matters tend to be worse whenever one extrapolates from
the present situation, which is perhaps known with satisfactory precision,
to future situations. The knowledge may not be accurate enough for a long-
term extrapolation. In addition to the fundamental lack of determinism in
quantum physics — the nondeterministic link between the state and the
phenomena — there is then a classical-type vagueness of the probabilistic
predictions, rather similar to the situation of classical deterministic chaos.
 Chapter 2

Kinematics:
How Quantum Systems are Described

2.1 Stern–Gerlach experiment

Now, turning to the systematic development of basic concepts and, at the

same time, of essential pieces of the mathematical formalism, let us con-
sider the historical Stern∗ –Gerlach† experiment of 1922. In a schematic
description of this experiment,

z
........... ...
... ................................................................................................... .................................
... ... .. .. ..
...... .....
y ...........................
... . .
.. ...
...
...
..............................................
.
beam of ... ... ..
....
...
...
...
...................................................................................................
........................................................... silver atoms ...
..
...
.. Ag..
..
................................
....
.
...
collimating oven
screen magnet apertures > 2000◦ C

(2.1.1)
silver atoms emerge from an oven (on the right), pass through collimat-
ing apertures, thereby forming a well-defined beam of atoms, and then
pass through an inhomogeneous magnetic field, eventually reaching a screen
where they are collected. The inhomogeneous field is stronger at the top
(z > 0 side) than the bottom (z < 0 side), which is perhaps best seen in a
frontal view:

atom beam mainly z .

.......... ..................................
at the center; ..
..
....
N.... .......
..... .....
corner shaped north pole
.. .....................
we are looking at ..
·
..... .. .. .. ..
..............................................
... ...
the approaching ... S
......................................
... flat south pole
atoms x .. . .
...........................
(2.1.2)

∗ Otto Stern (1888–1969) † Walther Gerlach (1889–1979)

9
10 Kinematics: How Quantum Systems are Described

Silver atoms are endowed with a permanent magnetic dipole moment so

that there is a potential energy

Emagn = −µ · B (2.1.3)

associated with dipole µ in the magnetic field B. It is the smallest if µ and

B are parallel and the largest when they are antiparallel:

B
........
B
........
B
.......
B
........
...... ...... ...... ......
...... ...... ...... ......
........ ......
...... ......
...... ...
µ ........
...... ..............
..............
.........
µ ........
......
......
........ ..
...... ...
...... ...
...... ... .......... ......... ...... ....
...... ... ..... ............ ...... ...
........ .... ........ ........ ........... . ........ ........
.......... .
......
......
........
......
........ ........... µ ...... ..
........
µ
...... ...... ...... ......

smallest, smaller, larger, largest potential energy.

(2.1.4)

As a consequence, there is a torque τ on the dipole exerted by the magnetic

field,

τ =µ×B, (2.1.5)

that tends to turn the dipole parallel to B. But since the intrinsic angular
momentum, the spin, of the atom is proportional to µ, and the rate of
change of the angular momentum is just the torque, we have
d
µ∝µ×B, (2.1.6)
dt
which is to say that the dipole moment µ precesses around the direction of
B, whereby the value of Emagn remains unchanged,
d dµ dB
Emagn = − ·B −µ· ∝ (µ × B) · B = 0 , (2.1.7)
dt dt dt
|{z}
=0
pictorially:

B
.......
......
......
........
......
......
......
......
........ .......
.......... ..............
...... ........
µ
........ (2.1.8)

In a homogeneous magnetic field, this is all that would be happening, but

the field of the Stern–Gerlach magnet is inhomogeneous: its strength de-
 Stern–Gerlach experiment 11

pends on position, mainly growing with increasing z. Therefore, the mag-

netic energy is position dependent,

Emagn (r ) = −µ · B(r ) , (2.1.9)

and that gives rise to a force F on the atom, equal to the negative gradient
of this magnetic energy,


F = −∇Emagn = ∇ µ · B(r ) . (2.1.10)

For µ · B > 0, the force is toward the region of stronger B field and for
µ · B < 0, it is toward the region of weaker field. Thus,

B
........
B
.......
B
........
B
........
B
........
...... ...... ...... ...... ......
...... ...... ...... ...... ......
...... .
........ .......... µ ......
...... ...............
.......
µ ......
......
......
........
...... .
........ ....
.......... ..... ..................... .......... .......... .......... .....
...... ...
........ ....
........
.......
...........................................
........
µ ...........
........ ..........
...... ..
........ ....
...... ........... ...... ..........
...... ...
..........
......
........
......
........
...... ........ ........ µ ........ µ
...... ........ ...... ......
.... .... .. .... ....

force upwards no force force downwards (2.1.11)

with the strength of the force proportional to the cosine of the angle between
µ and B because
B
.......
....... .
...... .. ... ... ..
....
........
θ ....
...
µ · B = µ B cos(θ) ......
........
...
..
(2.1.12)
...... ....
........ . .....
.
.
......
µ
.......... ..............
.................
.....

of course.
Now, the atoms emerging from the oven are completely unbiased in their
magnetic properties; there is nothing that would prefer one orientation of
µ and discriminate against others. All orientations are equally likely which
is to say that they will occur with equal frequency. Thus, some atoms
will experience a strong force upwards and others a weaker one, yet others
experience forces pulling downwards with a variety of strengths. Clearly,
then, we expect the beam of atoms to be spread out over the screen:
...
... .................................................................................................. .................................
... ..... . ... ... ..
...
... .............................................. beam of .. .. .. ....
... ....
...
...
...
...................................................................................................
........................................................... silver atoms ...
..
...
.. Ag
..
..
................................
....
.
...
collimating oven
screen magnet apertures
(2.1.13)
12 Kinematics: How Quantum Systems are Described

But this is not what is observed in such an experiment and was observed
by Stern and Gerlach in 1922. Rather, one finds just two spots:

...
... ................................................................................................... ................................
... ...... ... ... ... .
.. ....
...
...
...
.............................................. beam of ... ... .
....
...
...
...
....................................................................................................
......................................................... silver atoms ...
..
...
.. Ag
..
..
................................
....
.
...
collimating oven
screen magnet apertures
(2.1.14)
It is as if the atoms were prealigned with the magnetic field, some having
the dipole moment parallel and some having it antiparallel to B. But, of
course, there is no such prealignment, nothing prepares the atoms for this
particular geometry of the magnetic field. For, just as well, we could have
chosen the dominant field component in the x direction. Then, the splitting
into two would be along x, perpendicular to z, and the situation would be
as if the atoms were prealigned in the x direction. Clearly, the assumption
of prophetic prealignment is ludicrous.
Rather, we have to accept that the classical prediction of a spread-out
beam is inconsistent with the experimental observation. Classical physics
fails here, it cannot account for the outcome of the Stern–Gerlach experi-
ment.
Since there is nothing in the preparation of the beam that could possibly
bias the atoms toward a particular direction, we expect correctly that half of
the atoms are deflected up and half are deflected down. But an individual
atom is not split into two, an individual atom is either deflected up or
deflected down. And this happens in the perfectly probabilistic manner
discussed above. There is no possibility of predicting the fate of the next
atom if we think of performing the experiment in a one-atom-at-a-time
fashion.

2.2 Successive Stern–Gerlach measurements

Let us put a label on the atoms: We speak of a +atom when it is deflected

up (region z > 0) and of a −atom when it is deflected down (region z < 0).
In view of what we just noted, namely that we cannot possibly predict if
an atom will be deflected up or down, it may not be possible to attach
such labels. But, in fact, there is a well-defined sense to it. Consider atoms
that have been deflected up by the first Stern–Gerlach magnet and are then
passed through the second one:
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good woman, was slowly but surely unfolding in hers, and it reached
out toward her husband, it brooded over him in his sorrow and his
suffering, it was ready to forgive him, to lift him up. It had kindled in
her, too, the instinct of defense, the instinct to battle for those she
loved—as the leopardess in the jungle will battle to save her young.
The thought that the world was against him made him more than
ever her own. It was her portion now, not to fly from him, as she had
in the mountains, but to stand by him, to fight for him, to help him to
that moment—which she no longer doubted—the moment when he
should redeem himself, not only in her eyes, but in the eyes of the
world.
It seemed to her now, as she lay there, that these thoughts had been
with her through the night, that they had, indeed, possessed her with
a new gift, a kind of clairvoyance. She seemed to see into the mists
of the future and behold there—not the man who had failed at the
supreme test in the desert of ice and snow—but the soul of her
husband, purified by suffering and lifted to a courage greater than
death.
The room was shaded. The curtains, drawn the night before, still
shut out that feeble gleam of sunrise that shot down into the well-like
court of the city building. Diane turned her head on her pillow and
looked toward it, she could see a gleam of the sunlight striking, like
the golden head of an arrow, upon the dull wall opposite. It was the
herald of a new day—not only in the world, but in her life, the day
that she was to begin with her husband the greatest task of all, the
task of building up, of making his life over, of snatching back from
defeat and disgrace the career that he had chosen. That was the
thing she most keenly desired; he must not give up, he would not
now, she knew that. In the face of opposition, in the very teeth of
scandal, he would make good.
She rose slowly and went to the window. Looking up through the
half-drawn shutters she saw the sky, perfect and radiant and
ineffable. It lifted her heart, it reassured her. She began to dress
hastily, suddenly aware that she was late. Then she heard voices,
Arthur must have an early caller, she had been caught napping. She
hurried, half aware that the voices drew nearer, as if the speakers
had entered the room next her own. Then she was startled, she
recognized the voice which answered her husband—it was
Overton’s!
For a moment it gave her a shock, it was still impossible to ignore
that instant of emotion when they had stood together in the golden
mist of the rain, and her heart throbbed at the thought that Overton
must have believed that she had left her husband only for him. He
had a right to believe it! A deep blush rose to Diane’s brow and she
stood, wholly dressed now and ready to go to breakfast, but unable
to move. After last night it seemed strange to her that she could ever
have ignored the natural and spiritual law which bound her to Arthur.
Something had changed in her heart, or a new and deeper emotion,
an instinct as old as the world, had stirred within her. Was it that, was
it because—for the first time—she began to realize the dawn of a
new experience, of a tenderness so deep and so vital that it had
sanctified the bond between them, that she could no longer even
imagine the thought of deserting her husband? It might be that she
no longer tried to fathom it, but it was strong enough to steady her
now, she could go and meet Overton again without the emotion of
yesterday. To-day she was Arthur’s wife—beyond that there was
nothing!
She had taken a step toward the door and stopped, arrested by the
thought that the two men might have something to say to each other
about Arthur’s confession that they would not want her to hear. She
hesitated; there was nothing that she could not hear now, for her
husband had told her all. Yet——?
She was still standing there, when there was a soft knock at the door
and Faunce entered. His face was slightly flushed and his eyes
shone, but there was behind that a certain new strength that
reassured her. He came in quietly, and closed the door behind him.
“Diane,” he said in a low voice, “Overton is here. He’s come to tell
me something which seems—well, it seems almost unbelievable
after yesterday——” he paused and his flush deepened, but his eyes
held hers steadily. “He’s been sent—by the very men to whom I
confessed yesterday—to offer me the supreme command of the
expedition. He has finally refused it.”
For a moment Diane was unable to speak. The thought that the
chance had come to him—come at the moment when she had
seemed to foresee it—sent a thrill of joy through her. It was, indeed,
almost unbelievable. In the visions of the night, in her half waking
dreams, her very soul had cried out for this chance for him—and that
supreme but invisible Power who orders the fates of men had
answered her! She did not move, she stood still. With a half groping
gesture she put out her hand and Faunce took it, holding it close.
They said nothing, but he understood her, he knew that this, this
chance of redemption, had been the one desire of her heart.
“There’s one thing more, Diane,” he said softly, “Overton has told the
newspapers that he asked me to go, that he’s not strong enough yet
to assume command of an expedition. He wants to convince them
that my conduct wasn’t criminal, he has faced the terrors of ice and
snow and he knows—as I do—the terrible chance that both might be
lost when only one could be saved. He wants them to understand
that we still stand as friends, that he—he hasn’t condemned me as
the papers did last night! He’s done again the noble thing, the
expedition is to be mine, the chance is to be mine—to show you
——” his voice broke a little, but he smiled—“that your husband is no
longer a coward, that he’d rather die than to fail you again!”
Still she said nothing, but her hand quivered in his and he saw that
her dark lashes were wet with tears. There was no longer even a
shadow of doubt between them, he drew her slightly toward him,
watching her beautiful downcast face.
“I came to ask you,” he said quietly. “I’ll do nothing now that can
make you feel that I’m not willing to expiate, to make good. I came to
ask you, then, if I should take the command—after I gave up, take it
in the teeth of the clamor and the scandal? Take it—not as Overton’s
gift, but as my right, my right to earn my own chance to live or to die
doing my duty? Or would it nullify my expiation—must I suffer more?”
Again her hand quivered in his, but this time she lifted her eyes to
his, and he saw in them that new and exquisite tenderness, that
tranquillity which not even her tears could veil.
“I want you to go,” she replied softly. “I want it—because I have faith
in you, Arthur, I know that this time there is no power on earth that
can make you fail!”

In the days that followed, days in which the expedition was briefly
delayed while Faunce resumed his duties, he wrote to Gerry. Much
as he wanted Diane to go with him, he began to fear the hardships
for her. This new phase of their lives which was unfolding gradually
before their vision, made him anxious for her. Would it be well with
her if the child was born in that land of mist and snow? Could she
face the cold and the terrors, the possible hardships, even the
chance of privations? He said nothing of this to her, he knew her
longing to go, but he wrote to Gerry. Two days before the ship sailed
he received a letter from the doctor, and Diane received one from
her father.
The sight of his handwriting gave her a shock of mingled fear and
pleasure. Had he written to quarrel with her? It was not like him,
there was always too much finality about his rages. Or had he
relented? She remembered Overton’s words, that the judge would
forgive her. Did this mean that Overton had again intervened? Her
cheek reddened, but her eyes softened, after all, it was her father’s
way to do violent things violently. She opened the letter.
“Dear Diane,” the judge wrote; “Gerry has told me all that your
husband has written to him about you. Gerry and I are of one mind,
we can’t bear to have you face those hardships now. I said I’d
disown you. I’ve tried it, I can’t, you’re all I’ve got! I know how you
feel. Very well, I’ll forgive him, too. I’m down, I’m an old beggar alone
in the world. If I’m to have a grandchild I want it born in my house.
Will you come now, Diane, come to your old father?”
The letter rustled in her hands, she stood holding it and looking out
into the street. It was twilight, and one by one the lamps sprang up,
here and there and everywhere they twinkled and flashed and
danced, while long tiers of them on either side of the seemingly
endless street flashed and receded, light by light, until they
converged into a glow and brightness that made the hazy distance
seem like a spangled veil.
Diane was still standing there when Faunce rose from the table,
where he had read his letter, and came over to her side.
“Diane,” he said gently, “I wrote to Gerry, I told him. I’ve been afraid
the hardships were too great for you. Here’s his answer. He admits
the hardships, but he says you can face them if you will. You’re
young and strong. But still he wants you to stay, he wants to take
care of you himself.”
Diane turned quietly and gave him her father’s letter. She did not
look at him while he read it, for she knew he had suffered much at
her father’s hands, that she had been guilty of setting her father
against him. For the first time since that moment of confidence, of
complete reunion, she dreaded to look at him. Presently, however,
he handed it back to her and she met his eyes. They were calm, they
had, indeed, that new look of strength in them that nothing seemed
to dash. She knew the chloral habit had been absolutely broken, that
with a strength of will which amazed his doctor, he had let the drug
go. Now she saw that the moral change had been as great as the
physical.
“Will you stay?” he asked gently, his eyes holding hers.
She did not answer at once. It seemed as if she took that moment to
think, to concentrate all her powers of mind and heart on the one
supreme choice that was so vital to them both, the choice between
the risks and the hardships of the frozen pole and the safety of her
father’s house—without her husband. There was no question of a
quarrel now, the judge had forgiven him, he would stand by his word.
In his brusque way, Herford was holding out his hand to Faunce. To
go to him would not be an insult to her husband, but, if she left him
now, he must face the struggle alone and she had pledged herself to
face it with him. She had pledged herself, and she desired it more
than anything else in the world—except the safety of that little life
which might come in peril and cold and mist, like a pledge of their
faith to each other, and her belief that her husband would redeem
himself!
It seemed a long moment before she answered, and then, with a
mute, adorable gesture, she laid her cheek against his sleeve.
“I’m not afraid,” she said in her low, vibrating, beautiful voice, “I’m
going with you, Arthur.”
He made no answer in words, an inarticulate murmur was all that
escaped him. But he held her close and she seemed to feel the thrill
that her assurance gave him. She was no longer an outsider, no
longer a hostile critic at his fireside, they were united, their marriage
was no longer merely a physical, it was a spiritual union. Henceforth
she must share not only his victories, but his defeats, and in both, in
one as much as in the other, he would be dear to her, for she no
longer doubted him, she knew the worst that he had done, and she
knew, too, that he had repented and that now, purged by his long
spiritual conflict, he was in reality stronger than she was.
In the days which followed, days in which she wrote fully and lovingly
to her father, she was again conscious of a new and great
tranquillity. She had passed through the fiery furnace of her trial, she
had drained the cup of doubt to its dregs, and now she looked calmly
into that future that held for her the greatest of all trials, and the most
tender of all hopes.
The same thought was with her the day the ship sailed. It had been a
day of conflict for Faunce, a day of trial, for he had had to face the
publicity and the questions, but he had shown a strength and
composure that amazed himself. As he had told Diane, his
confession had freed him, he was no man’s slave, he had nothing to
fear, and he faced the future with a courage so high that it
transformed him. Diane saw it. She stood beside him as the ship,
slipping its moorings in the North River, dropped down the bay. It
was a day of clouds, and a light fog hung like a veil about the great
city, it made the distant streets appear like deep incisions between
the towering sky-scrapers, and the crowded battery was lightly
touched with mist. Above the gray clouds drifted, below the dark
water lapped, but Diane lifted her eyes to the face of her husband.
Faunce was calm; he was very pale but his eyes glowed and his lips
closed firmly. There was power in the face and conflict and hope.
Suddenly, the gray clouds parted and showed a rift of exquisite blue,
like a window in heaven, and a shaft of sunlight shot across the sky,
it touched the clouds with gold and it glinted on the towering figure of
Liberty bearing aloft her torch to light the world.
In the far distance the mists over the narrows grew soft and luminous
as Diane looked into them. She did not look back, she looked
forward. Out of that future, out of those clouds and that golden glory,
she seemed to see the form of her husband—no longer fallen and
defeated, but coming back to her in the semblance that she had
dreamed, clothed with powers at once mortal and spiritual, and
wearing the laurels of victory.
THE END
 TRANSCRIBER’S NOTES:
Obvious typographical errors have been corrected.
Inconsistencies in hyphenation have been
standardized.
Archaic or variant spelling has been retained.
 *** END OF THE PROJECT GUTENBERG EBOOK A CANDLE IN
THE WIND ***

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