1. Light is . . . Electrons are . . . Waves Initially thought to be waves
They do things waves do, like diffraction and interference
Wavelength – frequency relationship
Planck, Einstein, Compton showed us they behave like particles (photons)
Energy comes in chunks
Wave-particle duality: somehow, they behave like both
Photons also carry momentum
Momentum comes in chunks
Electrons are . . .
They act like particles
Energy, momentum, etc., come in chunks
They also behave quantum mechanically
Is it possible they have wave properties as well?

2. The de Broglie Hypothesis
Two equations that relate the particle-like and wave-like properties of light
1924 – Louis de Broglie postulated that these relationships apply to electrons as well
Implied that it applies to other particles as well
de Broglie could simply explain the Bohr quantization condition
Compare the wavelength of an electron in hydrogen to the circumference of its path
cancel 
Integer number of wavelengths fit around the orbit

3. Measuring wave properties of electrons
What energy electrons do we want?
For atomic separations, want distances around 0.3 nm  energies of 10 or so eV
How can we measure these wave properties?
Scatter off crystals, just like we did for X-rays!
Complication: electrons change speed inside crystal
Work function  increases kinetic energy in the crystal
Momentum increases in the crystal
Wavelength changes

4. The Davisson-Germer Experiment
Same experiment as scattering X-rays, except
Reflection probability from each layer greater
Interference effects are weaker
Momentum/wavelength is shifted inside the material
Equation for good scattering identical e-   d
Quantum effects are weird
Electron must scatter off of all layers

5. The Results: 1928: Electrons have both wave and particle properties
1900: Photons have both wave and particle properties
1930: Atoms have both wave and particle properties
1930: Molecules have both wave and particle properties
Neutrons have both wave and particle properties
Protons have both wave and particle properties
Everything has both wave and particle properties
Dr. Carlson has a mass of 82 kg and leaves this room at a velocity of about 1.3 m/s. What is his wavelength?

6. When wave-lengths are short, wave effects are hard to notice
Waves: How come we don’t notice?
Whenever waves encounter a barrier, they get diffracted, their direction changes
If the barrier is much larger then the waves, the waves change direction very little
If the barrier is much smaller then the waves, then the effect is enormous, and the wave diffracts a lot
When wave-lengths are short, wave effects are hard to notice
Light waves through a big hole
Sound waves through a small hole

7. Simple Waves  cos and sin have periodicity 2
If you increase kx by 2, wave will look the same
If you increase t by 2, wave will look the same
Simple waves look like cosines or sines:
k is called the wave number
Units of inverse meters
 is called the angular frequency
Units of inverse seconds
Wavelength  is how far you have to go in space before it repeats
Related to wave number k
Period T is how long you have to wait in time before it repeats
Related to angular frequency 
Frequency f is how many times per second it repeats
The reciprocal of period

8. Calculate the partial derivative below:
Math Interlude: Partial Derivatives
Ordinary derivatives are the local “slope” of a function of one variable f(x)
Partial derivatives are the local “slope” of a function of two or more variables f(x,y) in one particular direction
Partial derivatives are calculated the same way as ordinary derivatives, except other variables are treated as constant
Calculate the partial derivative below:

9. What is the dispersion relationship for light in vacuum?
Waves come about from the solution of differential equations
For example, for light
These equations lead to relationships between the angular frequency  and the wave number k
Called a dispersion relation
What is the dispersion relationship for light in vacuum?
Need to find a solution to wave equation, let’s try:

10. Phase velocity 
The wave moves a distance of one wavelength  in one period T
From this, we can calculate the phase velocity denoted vp
It is how fast the peaks and valleys move
What is the phase velocity for light in vacuum?
Not constant for most waves!

11. Adding two waves
Real waves are almost always combinations of multiple wavelengths
Average these two expressions to get a new wave:
This wave has two kinds of oscillations:
The oscillations at small scales
The “lumps” at large scales

12. Small scale oscillations Large scale oscillations
Analyzing the sum of two waves:
Need to derive some obscure trig identities:
Average these:
Substitute:
Rewrite wave function:
Small scale oscillations
Large scale oscillations

13. First hint of uncertainty principle
The “uncertainty” of two waves k1 k k k2 k
Our wave is made of two values of k:
k is the average value of these two
k is the amount by which the two values of k actually vary from k
The value of k is uncertain by an amount k
Each “lump” is spread out in space also
Define x as the distance from the center of a lump to the edge
The distance is where the cosine vanishes
Plotted at t = 0
First hint of uncertainty principle
x

14. Small scale oscillations Large scale oscillations
Group Velocity
Small scale oscillations
Large scale oscillations
The velocity of little oscillations governed by the first factor
Leads to the same formula as before for phase velocity:
The velocity of big oscillations governed by the second factor
Leads to a formula for group velocity:
These need not be the same!

15. More Waves One wave Two waves allow you to create localized “lumps”
Three waves allow you to start separating these lumps
More waves lets you get them farther and farther apart
Infinity waves allows you to make the other lumps disappear to infinity – you have one lump, or a wave packet
A single lump – a wave packet – looks and acts a lot like a particle
Two waves
Three waves
Five waves
Infinity waves

16. Wave Packets
We can combine many waves to separate a “lump” from its neighbors
With an infinite number of waves, we can make a wave packet
Contains continuum of wave numbers k
Resulting wave travels and mostly stays together, like a particle
Note both k-values and x-values have a spread k and x.

17. What is the phase and group velocity for this wave?
Compare to two wave formulas:
Phase velocity formula is exactly the same, except we simply rename the average values of k and  as simply k and 
Group velocity now involves very closely spaced values of k (and ), and therefore we rewrite the differences as . . .
What is the phase and group velocity for this wave?

18. What is the phase and group velocity for this wave?
Sample Problem
What is the phase and group velocity for this wave?
Moved 30 m
Finish, t = 30 s
Start, t = 0 s
Moved 60 m

19. What’s wrong with the following proof?
Phase and Group velocity
How to calculate them:
You need the dispersion relation: the relationship between  and k, with only constants in the formula
Example: light in vacuum has
What’s wrong with the following proof?
If the dispersion relation is  = Ak2, with A a constant, what are the phase and group velocity?
Theorem: Group velocity
always equal phase velocity
doesn’t

20. The Classical Uncertainty Principle
The wave number of a wave packet is not exactly one value
It can be approximated by giving the central value
And the uncertainty, the “standard deviation” from that value k k
The position of a wave packet is not exactly one value
It can be approximated by giving the central value
And the uncertainty, the “standard deviation” from that value x x
These quantities are related:
Typically, x k ~ 1
Precise Relation: (proof hard)

21. Uncertainty in the Time Domain
Stand and watch a wave go by at one place
You will see the wave over a period of time t
You will see the wave with a combination of angular frequencies 
The same uncertainty relationship applies in this domain

22. Estimating Uncertainty: Carlson’s Rule
A particle/wave is trapped in a box of size L
What is the uncertainty in its position x? L ?
Guess of position
L/2
L/2
Best guess: The particle is in the center, x = L/2
But there is an error x on this amount
It is no greater than L/2
It is certainly bigger than 0
Carlson’s rule: use x = L/4
This rule can be applied in the time domain as well
Exact numbers for x:
Particle in a box: 0.181L
Uniform distribution: 0.289L

23. Sample Problem:
A student is supposed to measure the frequency of an object vibrating at f = Hz, but he’s late for his next class, so he only spends s gathering data. How much error is he likely to have due to his hasty data sampling?
Since the data was taken during s, the date fits into a time box of length s
By Carlson’s rule, we have t = s
By the uncertainty principle (time domain), we have:
Since f = /2, this causes an estimated error of
Of course, the error could be much larger than this

24. Wave Equations You Need:
These equations always apply
Two equations describing a generic wave
Light waves only

25. What’s the imaginary part of 4 + 7i?
Math Interlude: Complex Numbers
A complex number z is a number of the form z = x + iy, where x and y are real numbers and i = (-1).
x is called the real part of z and y is called the imaginary part of z.
The complex conjugate of z, denoted z* is the same number except the sign of the imaginary part is changed
What’s the imaginary part of 4 + 7i?
Note: no i
Adding, subtracting, and multiplying complex numbers is pretty easy:
To divide complex numbers, multiply numerator and denominator by the complex conjugate of the denominator

26. A Useful Identity Taylor series expansion
Apply to sin, cos, and ex functions
In last expression, let x  i

27. What is the dispersion relationship for light in vacuum?
Complex Waves
Typical waves look like:
We’d like to think about them both at once
We’d like to make partial derivatives as simple as possible
A mathematical trick lets us achieve both goals simultaneously:
Real part is cosine
Imaginary part is sine
This makes the derivatives easier in differential equations:
What is the dispersion relationship for light in vacuum?

28. Magnitudes of complex numbers
The magnitude of a complex number z = x + iy denoted |z|, is given by:
This formula is rarely used
The square of the magnitude can be written
This is the easiest way to calculate it

29. What’s the magnitude squared of the following expression?
Sample Problem
What’s the magnitude squared of the following expression?