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?

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

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

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

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?

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

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

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:

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:

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!

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

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

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

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!

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

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.

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?

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

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

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)

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

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

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

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

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

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

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?

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

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