Light is . . .

1. Light is . . . 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 cancel  • Two equations that relate the particle-like and wave-like properties of light • 1924 – Louis de Broglie postulated that theserelationships 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 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. 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. 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. Dispersion Relations • 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. Analyzing the sum of two waves: Large scale oscillations Small scale oscillations • Need to derive some obscure trig identities: • Average these: • Substitute: Rewrite wave function:

13. The “uncertainty” of two waves k1 k2 k k 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. 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. Phase and Group velocity • 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. Sample Problem Moved 30 m Moved 60 m What is the phase and group velocity for this wave? Finish, t = 30 s Start, t = 0 s

19. 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 Guess of position L/4 • A particle/wave is trapped in a box of size L • What is the uncertainty in its position x? L ? 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 = 147.0 Hz, but he’s late for his next class, so he only spends 0.100 s gathering data. How much error is he likely to have due to his hasty data sampling? • Since the data was taken during 0.100 s, the date fits into a time box of length 0.100 s • By Carlson’s rule, we have t = 0.0250 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. Math Interlude: Complex Numbers Note: no i • A complex numberz 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? • 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 exfunctions In last expression, let x i

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