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CHAPTER 4
Wave Properties of Particles
4.1
4.2
4.3
4.4
4.6
4.8
4.7
X-Ray Scattering
De Broglie Waves
Electron Scattering
Wave Motion
Uncertainty Principle
Particle in a Box
Probability, Wave Functions, and
the Copenhagen Interpretation
4.5 Waves or Particles?
Louis de Broglie
(1892-1987)
I thus arrived at the overall concept which guided my studies: for both
matter and radiations, light in particular, it is necessary to introduce the
corpuscle concept and the wave concept at the same time.
- Louis de Broglie, 1929
Electron Scattering
George P. Thomson (1892–1975),
son of J. J. Thomson, saw electron
diffraction from celluloid, gold,
aluminum, and platinum.
A randomly oriented
polycrystalline sample of SnO2
produces rings.
In 1925, Davisson and
Germer observed
electrons diffracting
(much like x-rays) from
nickel crystals.
Beautiful Proof That Electrons are
Waves: Imaging Using Them
Imaging using
light waves is
well known.
But optical
microscopes’
resolution is
only l/2 ~
200nm.
Electron micrograph of pollen grains with ~0.1nm
resolution
Electrons have
much smaller
wavelengths,
and electron
microscopes
can achieve
resolutions of
~0.05nm.
Recall that waves diffract through slits.
Fraunhofer diffraction patterns
One slit
Two slits
In 1803, Thomas Young saw the two-slit pattern for light,
confirming the wave nature of light. But particles are also
waves. So they should exhibit similar patterns when passing
through slits, especially pairs of slits.
Electron Double-Slit Experiment
C. Jönsson of Tübingen,
Germany, succeeded in 1961
in showing double-slit
interference effects for
electrons by constructing very
narrow slits and using
relatively large distances
between the slits and the
observation screen.
This experiment demonstrated
that precisely the same
behavior occurs for both light
(waves) and electrons
(particles).
Which slit does an
electron go through?
Shine light on the double slit and observe with a microscope.
This should tell us which slit the electron went through.
The photon momentum:
p ph 
h
l ph
h
h

d
h

The electron momentum: pel 
lel d
Need lph < d (the slit
separation) to distinguish the
slits.
Diffraction is significant only
when the slit separation d ≤ lel
the wavelength of the e wave.
So the photon momentum pph > h/d ≥ pel . It’s enough to strongly
modify the momentum of the electron, strongly deflecting it!
The attempt to identify which slit the electron passes through
changes the diffraction pattern, washing out the fringes!
So we can’t tell which slit the electron went through.
Which slit does a photon go through?
Dimming the light in Young’s two-slit experiment results in single
photons at the screen. Since photons are particles, each can only
go through one slit. So, at such low intensities, their distribution
should become the single-slit pattern.
Each photon
actually goes
through both
slits!
x
Two-Slit Experiment with
Single Electrons
The same is true for
electrons!
Wave-particle-duality solution
It’s very confusing that everything is both a particle and a wave.
The wave-particle duality is a little less confusing if we think in terms
of:
Bohr’s Principle of Complementarity: It’s not possible to describe
physical observables simultaneously in terms of both particles and
waves.
When we’re making a measurement, use the particle description, but
when we’re not, use the wave description.
When we’re looking, fundamental quantities are particles; when
we’re not, they’re waves.
In the two-slit problem, the electrons propagate as waves but are
detected as particles.
Uncertainty Principle:
Energy Uncertainty
The energy uncertainty of a wave packet is:
E  h   h

 
2
In the Uncertainty
Principle, we’ll
henceforth use a
width definition that
yields an
uncertainty product
of ½.
Combined with the angular frequency
relation we derived earlier:
 t 
E
1
t 
h
2
Energy-Time Uncertainty Principle:
E t  h / 2
Werner Heisenberg (1901–1976)
Momentum Uncertainty Principle
The same mathematics relates x and k:
k x ≥ ½
So it’s also impossible to measure simultaneously
the precise values of k and x for a wave.
Now the momentum can be written in terms of k:
h
h
p 
 (h / 2 )k
l 2 / k

So the uncertainty in momentum is:
p  hk
p  h  k
h
But multiplying k x ≥ ½ by ħ: h k x 
2
And we have Heisenberg’s Uncertainty Principle:
px x 
2
How to think about Uncertainty
The act of making one measurement perturbs the other.
Precisely measuring the time disturbs the energy.
Precisely measuring the position disturbs the momentum.
The Heisenberg-mobile. The problem was that when
you looked at the speedometer you got lost.
Kinetic Energy Minimum
Since we’re always uncertain as to the exact position, x  ,
of a particle, for example, an electron somewhere inside an
atom, the particle can’t have zero kinetic energy:
/2
p 

x 2
The average of a positive quantity must always equal or
exceed its uncertainty:
pave
/2
 p 

x 2
so:
K ave
2
2
pave
(p) 2



2m
2m
8m 2
Wave Motion
De Broglie matter waves should be described in a manner
similar to light waves. The matter wave should also be a
solution to a wave equation.
And it will often have a solution like:
Y(x,t) = A exp[i(kx – t – q)]
x
Define the wave number
k and the angular
frequency  as usual:
k
2
l
and
2

T
Probability, Wave Functions, and the
Copenhagen Interpretation
Okay, if particles are also waves, what’s waving?
Probability
The wave function determines the likelihood (or probability) of
finding a particle at a particular position in space at a given time:
P( x)  Y ( x)
2
The probability of the
particle being
between x1 and x2 is
given by:
x2

Y ( x) dx
2
x1



Y ( x) dx  1
2
The total probability of finding the
particle is 1. Forcing this condition on
the wave function is called
normalization.
The Copenhagen Interpretation
1. A system is completely described by a wave function Y, which
represents an observer's knowledge of the system. (Heisenberg)
2. The description of nature is probabilistic. The probability of an event is
the mag squared of the wave function related to it. (Max Born)
3. Heisenberg's Uncertainty Principle says it’s impossible to know the
values of all of the properties of the system at the same time;
properties not known with precision are described by probabilities.
4. Complementarity Principle: matter exhibits a wave-particle duality. An
experiment can show the particle-like properties of matter, or wave-like
properties, but not both at the same time. (Bohr)
5. Measuring devices are essentially classical devices, and they measure
classical properties such as position and momentum.
6. The correspondence principle of Bohr and Heisenberg: the quantum
mechanical description of large systems should closely approximate
the classical description.
Particle in a Box
A particle (wave) of mass m is in a one-dimensional
box of width ℓ.
The box puts boundary conditions on the wave. The
wave function must be zero at the walls of the box
and on the outside.
In order for the probability to vanish at the walls, we
must have an integral number of half wavelengths in
the box:
nl

2
The energy:
or
ln 
2
n
(n  1, 2,3,...)
2
2
p
h
E  K  12 mv2 

2m 2ml 2
The possible wavelengths
are quantized and hence
so are the energies:
Probability of the particle vs. position
Note that E0 = 0 is not a
possible energy level.
The concept of energy
levels, as first discussed in
the Bohr model, has
surfaced in a natural way
by using waves.
The probability of
observing the particle
between x and x + dx in
each state is:
P( x)  Y ( x)
2
Bohr’s Quantization Condition revisited
One of Bohr’s assumptions in his hydrogen atom model was that the
angular momentum of the electron in a stationary state is nħ.
This turns out to be equivalent to
saying that the electron’s orbit
consists of an integral number
of electron de Broglie
wavelengths:
nh
L  rp 
 nh
2
Multiplying by 2/p, we find the
circumference:
nh
2 r 
 nl
p
Circumference
electron
de Broglie
wavelengt
h