WAVE PARTICLE DUALITY

WAVE PARTICLE DUALITY

Evidence for wave-particle duality

• Photoelectric effect
• Compton effect

• Electron diffraction
• Interference of matter-waves

Consequence: Heisenberg uncertainty principle

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 WAVE PARTICLE DUALITY
Hertz J.J. Thomson
PHOTOELECTRIC EFFECT
When UV light is shone on a metal plate in a vacuum,
it emits charged particles (Hertz 1887), which were
later shown to be electrons by J.J. Thomson (1899).

Vacuum Light,
frequency ν Classical expectations
chamber
Metal Collecting Electric field E of light exerts
plate plate force F=-eE on electrons. As
intensity of light increases, force
increases, so KE of ejected
electrons should increase.
Electrons should be emitted
whatever the frequency ν of the
I light, so long as E is sufficiently
Ammeter large
For very low intensities, expect a
Potentiostat time lag between light exposure
and emission, while electrons
absorb enough energy to escape
from material
 WAVE PARTICLE DUALITY
PHOTOELECTRIC
Actual results:
EFFECT
Einstein’s
(cont) Einstein

interpretation
Maximum KE of ejected (1905):
electrons is independent of
intensity, but dependent on ν Light comes in
packets of energy
For ν<ν0 (i.e. for frequencies
below a cut-off frequency) no
(photons) E  h  Millikan
electrons are emitted An electron
There is no time lag. absorbs a single
However, rate of ejection of photon to leave
electrons depends on light the material
intensity.

The maximum KE of an emitted electron is then

Kmax  h  W
Work function: minimum Verified in
Planck constant: energy needed for electron detail through
universal constant to escape from metal subsequent
of nature (depends on material, but experiments by
h  6.63 1034 Js usually 2-5eV) Millikan
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SUMMARY OF PHOTON PROPERTIES
Relation between particle and wave properties of light
Energy and frequency E  h
Also have relation between momentum and wavelength
Relativistic formula relating
energy and momentum E 2  p 2c 2  m2c 4
For light E  pc and c  
h
h
p 
 c
Also commonly write these as wavevector
2 h
E   p  k   2 k 
angular frequency  hbar 2 LoJ
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Compton
COMPTON SCATTERING
Compton (1923) measured intensity of scattered X-
rays from solid target, as function of wavelength for
different angles. He won the 1927 Nobel prize.
Collimator Crystal
X-ray source
(selects angle) (measure
wavelenght)

θ
Target

Result: peak in scattered Detector

radiation shifts to longer
wavelength than source. Amount
depends on θ (but not on the
target material).
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Classical picture: oscillating electromagnetic field causes oscillations in

positions of charged particles, which re-radiate in all directions at same
frequency and wavelength as incident radiation.
Change in wavelength of scattered light is completely unexpected
classically

Incident light wave Oscillating Emitted light wave

electron
Compton’s explanation: “billiard ball” collisions between
particles of light (X-ray photons) and electrons in the material

Before After p 
Incoming
scattered photon
photon
θ
p
Electron LoJ

pe scattered electron
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Before After p 
Incoming scattered photon
photon θ
p
Electro
n pe scattered electron

Conservation of energy Conservation of momentum

h  me c  h    p c  m c 
2 4 1/ 2 hˆ
2 2 2
p  i  p   pe

e e

From this Compton derived the change in wavelength

h
    1  cos 
me c
 c 1  cos    0
h
c  Compton wavelength   2.4 1012 m
me c LoJ
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Note that, at all angles

there is also an unshifted peak.

This comes from a collision

between the X-ray photon and
the nucleus of the atom
h
    1  cos   > 0
mN c

since mN > me

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WAVE-PARTICLE DUALITY OF LIGHT

In 1924 Einstein wrote:- “ There are therefore now two theories
of light, both indispensable, and … without any logical connection.”
Evidence for wave-nature of light
• Diffraction and interference
Evidence for particle-nature of light
• Photoelectric effect
• Compton effect
•Light exhibits diffraction and interference
phenomena that are only explicable in terms of
wave properties
•Light is always detected as packets (photons); if
we look, we never observe half a photon
•Number of photons proportional to energy density
(i.e. to square of electromagnetic field strength)
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MATTER WAVES De Broglie

We have seen that light comes in discrete units (photons) with

particle properties (energy and momentum) that are related to the
wave-like properties of frequency and wavelength.

In 1923 Prince Louis de Broglie postulated that ordinary matter can have
wave-like properties, with the wavelength λ related to momentum
p in the same way as for light

de Broglie relation h Planck‟s constant

 h  6.63 1034 Js
de Broglie wavelength
p
wavelength depends on momentum, not on the physical size of the particle
Prediction: We should see diffraction and interference
of matter waves LoJ
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Estimate some de Broglie wavelengths
• Wavelength of electron with 50eV kinetic
energy 2 2
p h h
K      1.7  10 10
m
2me 2me  2
2me K

• Wavelength of Nitrogen molecule at room temp.

3kT
K , Mass  28m u
2
h
  2.8 1011 m
3MkT
• Wavelength of Rubidium(87) atom at 50nK
h
  1.2 106 m
3MkT
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ELECTRON DIFFRACTION: The Davisson-Germer experiment (1927)

G.P.
Davisson Thomson
The Davisson-Germer
θi experiment: scattering a beam
of electrons from a Ni crystal.
Davisson got the 1937 Nobel
prize.
θi
At fixed angle, find sharp peaks in
intensity as a function of electron
energy
Davisson, C. J.,
At fixed accelerating voltage "Are Electrons
(fixed electron energy) find a Waves?,"
pattern of sharp reflected beams Franklin
from the crystal Institute
Journal 205,
597 (1928)
G.P. Thomson performed similar
interference experiments with thin-
film samples LoJ
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ELECTRON DIFFRACTION (cont)

Interpretation: similar to Bragg scattering of X-rays from
crystals

θi Path
a cos i difference:
a(cos r  cos i )
θr
Constructive interference
a
when
a(cos r  cos i )  n

Electron scattering
dominated by
surface layers a cos r Note difference from usual “Bragg‟s
Law” geometry: the identical
Note θi and θr not scattering planes are oriented
necessarily equal perpendicular to the surface
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THE DOUBLE-SLIT EXPERIMENT

Originally performed by Young (1801) to demonstrate the wave-nature
of light. Has now been done with electrons, neutrons, He atoms among
others.

Alternative
method of
y detection: scan
a detector
across the
d
θ plane and
record number
Incoming d sin  of arrivals at
coherent beam each point
of particles (or Detecting
light) screen
D
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For particles we expect two peaks, for waves an interference pattern

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Neutrons, A
Zeilinger et al.
1988 Reviews of
Modern Physics 60
1067-1073

He atoms: O Carnal and J

Mlynek 1991 Physical Review
Letters 66 2689-2692
C60 molecules: M
Arndt et al. 1999 Fringe
Nature 401 680- visibility
682
decreases as
With multiple- molecules
slit grating
are heated.
L.
Hackermülle
Without grating r et al. 2004
Nature 427
711-714
Interference patterns can not be explained classically - clear demonstration of matter waves
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DOUBLE-SLIT EXPERIMENT WITH HELIUM ATOMS
(Carnal & Mlynek, 1991,Phys.Rev.Lett.,66,p2689)
Path d sin 
difference:
Constructive interference: d sin   n
D y
Separation between maxima: y 
(proof following) d
Experiment: He atoms at 83K, d
with d=8μm and D=64cm θ
d sin 
Measured separation:y  8.2 m
Predicted de Broglie
wavelength: D
3kT
K , Mass  4m u
2 Predicted separation:y  8.4  0.8 m
h
  1.03 1010 m Good agreement with experiment
3MkT LoJ
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FRINGE SPACING IN DOUBLE-SLIT EXPERIMENT
Maxima when: d sin   n
D d so use small angle approximation
n
 y
d

   d
d θ

d sin 
Position on screen:y  D tan   D

So separation between adjacent

D
maxima:
y  D
D
 y 
d
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DOUBLE-SLIT EXPERIMENT INTERPRETATION
• The flux of particles arriving at the slits can be reduced so that only
one particle arrives at a time. Interference fringes are still observed!
Wave-behaviour can be shown by a single atom.
Each particle goes through both slits at once.
A matter wave can interfere with itself.
Hence matter-waves are distinct from H2O molecules collectively
giving rise to water waves.
• Wavelength of matter wave unconnected to any internal size of
particle. Instead it is determined by the momentum.
• If we try to find out which slit the particle goes through the
interference pattern vanishes!
We cannot see the wave/particle nature at the same time.
If we know which path the particle takes, we lose the fringes .

The importance of the two-slit experiment has been memorably summarized

by Richard Feynman: “…a phenomenon which is impossible, absolutely impossible,
to explain in any classical way, and which has in it the heart of quantum mechanics.
In reality it contains the only mystery.”
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HEISENBERG MICROSCOPE AND

THE UNCERTAINTY PRINCIPLE
(also called the Bohr microscope, but the thought
experiment is mainly due to Heisenberg).
The microscope is an imaginary device to measure
Heisenberg
the position (y) and momentum (p) of a particle.

Particle
θ/2
y
Light source,
wavelength λ
Resolving power of lens:
Lens, with angular
diameter θ 
y 

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HEISENBERG MICROSCOPE (cont)
Photons transfer momentum to the particle when they scatter.
Magnitude of p is the same before and after the collision. Why?
p
Uncertainty in photon y-momentum
= Uncertainty in particle y-momentum
θ/
 p sin  / 2   p y  p sin  / 2  2
p
Small angle approximation
p y  2 p sin  / 2   p
h
de Broglie relation givesp  h /  and so p y 


From before y  hence p y y  h

HEISENBERG UNCERTAINTY PRINCIPLE.
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Point for discussion

The thought experiment seems to imply that,
while prior to experiment we have well defined
values, it is the act of measurement which
introduces the uncertainty by disturbing the
particle‟s position and momentum.

Nowadays it is more widely accepted that quantum

uncertainty (lack of determinism) is intrinsic to
the theory.

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HEISENBERG UNCERTAINTY PRINCIPLE
xpx  / 2
yp y  / 2
zpz  / 2

HEISENBERG UNCERTAINTY PRINCIPLE.

We cannot have simultaneous knowledge

of „conjugate‟ variables such as position and momenta.

Note, however, xp y  0 etc

Arbitrary precision is possible in principle for

position in one direction and momentum in another
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HEISENBERG UNCERTAINTY PRINCIPLE
There is also an energy-time uncertainty relation

Et  / 2
Transitions between energy levels of atoms are not perfectly
sharp in frequency.

n=3 An electron in n = 3 will spontaneously

E  h 32 decay to a lower level after a lifetime
n=2
of ordert 108 s

n=1
Intensity

There is a corresponding „spread‟ in

 32
the emitted frequency

 32 Frequency
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CONCLUSIONS
Light and matter exhibit wave-particle duality

Relation between wave and particle properties

given by the de Broglie relations h
E  h p
, 
Evidence for particle properties of light
Photoelectric effect, Compton scattering

Evidence for wave properties of matter

Electron diffraction, interference of matter waves
(electrons, neutrons, He atoms, C60 molecules)
xpx  / 2
Heisenberg uncertainty principle limits
simultaneous knowledge of conjugate variables yp y  / 2
zpz  / 2 LoJ