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):
For light : E
 h
 and p

E c
 h
 c
 h
 deBroglie :
For the electron :
  h p
 h mv connection between " particle like" behavior and " wave like" behavior
Connection to the Bohr Model:
For constructive interference:
2
 r
 n

2
 r
 n h mv mvr
 n h
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 1x10
3
eV x-rays is

= hc/E = 1240 eV nm / (1x 10
3
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.

E
 h p
 eV
 h
2 mE because E

1
2 mv
2  where V is the accelerati ng voltage p
2
2 m p

2 mE
  h
2 meV
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:
( )
  
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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