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Physics 200 Class #18 Notes
November 7, 2005
There are five major topics in this set of notes
 deBroglie Waves
 Electron diffraction
 A wave equation
 The Pauli Principle
 A particle in a box
 Probability Interpretation
Central problem of the early 1920s: Make the Bohr atom work for heavier atoms.
deBroglie Waves-the electron as a wave
deBroglie’s hypothesis (1924):
E h h
For light : E  h
and
p 

c
c

deBroglie :
h
h
For the electron :
 
connection between " particle - like" behavior
p mv
and " wave - like" behavior
Connection to the Bohr Model:
For constructive interference:
2 r  n
h
2 r  n
mv
h
mvr  n
2
and mvr is called the angular momentum
Important point - not only is energy and position quantized, the angular momentum is too.
When integers appear (the "n" tells you to plug in 0, 1, 2, etc) that's quantum mechanics!
A pictorial representation:
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and a simple demonstration: Standing waves on a circular wire can form when the
circumference is equal to a whole number of wavelengths.
Electron diffraction
Can the electron wave be detected? The answer is supplied by the Davisson-Germer
experiment at Bell Labs! They looked at crystal diffraction patterns using electrons.
First, what is crystal diffraction?
Light rays reflecting off the different faces of the crystal can interfere with each other. This
is similar to thin film interference we spoke about. In the diagram below, we're looking at
the path length difference between rays bouncing off the top layer of a crystal and the second
layer. The important thing about this, you get constructive interference at certain angles,
which tells you about the structure of the crystal.
For a sense of scale, the wavelength of 1x103 eV x-rays is
 = hc/E = 1240 eV nm / (1x 103 eV) = 1 nm. That's 500 times smaller than most visible
light.
This can also be applied to more complicated molecules. Below you see what happens when
you scatter x-rays off of DNA. The x-rays are hitting the strand at the very end. The pattern
you see is what first led researchers to believe that DNA is a double helix. The image below
is one of the actual images that they looked at. The diffraction pattern was analyzed by
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Rosalind Franklin, though Crick and Watson got the Noble prize for it. When the caption
below states that the diffraction analysis "experimentally confirmed the then current guesses
that DNA was helical" it is misleading. The reason why there even were "current guesses"
was because Watson and Crick looked at her lab books. It's an interesting story to look into.
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We see that x-rays can be diffracted off of crystals. And we see that de Broglie thought of
electrons as behaving like waves. The next step was to look for diffraction of electrons to see
if they really could behave like waves.
Electron Diffraction!!!!!!!!!!!!!!
Electron Diffraction can also be done with double slits-we return to this later.
We can reproduce this famous experiment by sending an electron beam from a TV tube onto
a collection of single crystals of Aluminum that are randomly oriented. Because of the
random orientation, the constructive interference pattern should be concentric circles.
h
h
p2
2
1
 
because E  2 mv 
p  2mE
p
2m
2mE
E  eV where V is the accelerati ng voltage
h

2meV
As V is increased, the wavelength is decreased and the pattern should shrink.
Just accelerating the electrons with 100 Volts (about what comes out of the power socket in
your wall) the wavelength of the electrons is less than 0.1 nm. For a sense of scale, that's
5,000 times smaller than visible light. If you accelerate the electrons with 10,000 Volts you
get a 0.01 nm wavelength! That's 10 times smaller than a hydrogen atom!
What else behaves like waves?
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Neutron Diffraction!!!!!!!
Look at the wavelength equations for the electron. The mass of a neutron is 2,000 times
bigger than the electron. For the electron accelerated with 10,000 Volts, moving pretty fast,
the wavelength is 0.01 nm, or 1 x 10 -11 m. For a neutron moving the same speed, the
wavelength would be 0.0002 nm or 2 x 10 -13 m. That's a million times smaller wavelength
than visible light!
If light behaves like a wave, and electrons do too, and so do neutrons, we come to the
conclusion that...
EVERYTHING IS WAVELIKE!!! Every particle behaves like a wave too! Even big things
like you.
If we look again at the two slit experiment:
Young’s Experiment with Electrons
Same results as for photons!
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Single-electron events build up over a 20 minute exposure to form an interference pattern in this
double-slit experiment by Akira Tonomura and co-workers. (a) 8 electrons; (b) 270 electrons; (c)
2000 electrons; (d) 60,000. A video of this experiment will soon be available on the web
(www.hqrd.hitachi.co.jp/em/doubleslit.html).
Source of controversy: Quantum theory only assigns probabilities to possible outcomes of an
experiment.
Probability of detecting an electron in a small region x (between x and x  x ) is equal to:
P( x)x   2 x
Bohr: By an act of measurement we push nature into giving us one answer or another. Thus if
we include the measuring procedure and apparatus in the description of the physical
situation, all will be well.
Bohr comes to a line of reasoning known as the Copenhagen Interpretation of Quantum
Theory. The wave function describes all of the possible outcomes of an experiment. In a
measurement, one of these possibilities becomes known. When does this collapse of the
wave function to the measured position take place? The Copenhagen answer is that it occurs
when the experimenter becomes aware of the result of the measurement!
We said that light, or electrons, or whatever, travels like a wave and arrives like a particle.
We only have a probability that the electron will land in any particular place. After a while
we can see the higher probability places have more electrons landing in them and we get
interference patterns. If we actually try to measure where the electrons are when they are
traveling, they no longer travel like waves. The act of measuring them makes them behave
like a particle and you can no longer get interference!
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