Light is . . . •Initially thought to be waves •They do things waves do, like diffraction and interference •Wavelength – frequency relationship cf •Planck, Einstein, Compton showed us they behave like particles (photons) •Energy comes in chunks •Wave-particle duality: somehow, they behave like both E hf •Photons also carry momentum •Momentum comes in chunks p E c hf c h 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? p h The de Broglie Hypothesis •Two equations that relate the particle-like and wave-like properties of light E hf p h 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 L n mevr pr hr 2 r cancel n 2 r C Integer number of wavelengths fit around the orbit Measuring wave properties of electrons •What energy electrons do we want? p h 4.136 10 eV s 3.00 10 m/s p hc E mv 2 2 2m 2mc 2 0.511106 eV 2 2 2 15 2 2 8 2 2 1 2 1.504 10 18 2 eV m 2 nm 1.504 eV 2 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- 2d cos m 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? h h 6.626 1034 J s 6.22 1036 m p mv 82 kg 1.3 m/s 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 Light waves through a big hole When wavelengths are short, wave effects are hard to notice l 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 x, t A cos kx t •Units of inverse meters • is called the angular frequency x, t A sin kx t •Units of inverse seconds •Wavelength is how far you have to go in space before it repeats 2 k •Related to wave number k •Period T is how long you have to wait in time before it repeats •Related to angular frequency 2 T 2 f •Frequency f is how many times per second it repeats •The reciprocal of period Math Interlude: Partial Derivatives f x h f x •Ordinary derivatives are the local “slope” of a d f x lim function of one variable f(x) h 0 dx h •Partial derivatives are the local “slope” f x, y f x h, y f x, y of a function of two or more variables lim f(x,y) in one particular direction h 0 x h •Partial derivatives are calculated the same way as ordinary derivatives, except other variables are treated as constant d Ax2 B Ax 2 B d 2 Ax2 B e e Ax B 2 Axe dx dx Ax2 Ay 2 e Ax2 Ay 2 Ax 2 Ay 2 Ax 2 Ay 2 2 Axe e x x Calculate the partial derivative below: cos kx t sin kx t kx t k sin kx t x x Dispersion Relations E 2 E 2 2 z z •Waves come about from the solution of differential equations c 0 2 2 t x •For example, for light •These equations lead to relationships between the angular frequency and the wave number k 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: E A cos kx t Ez kA sin kx t x 2 Ez 2 k A cos kx t 2 x z Ez A sin kx t t 2 Ez 2 A cos kx t 2 t 2 c2k 2 2 A cos kx t c2k 2 A cos kx t 0 ck Phase velocity 1 f T 2 k 2 vp f k •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 f 2 vp T k 2 k What is the phase velocity for light in vacuum? ck c vp k k 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: 1 cos k1 x 1t x, t 12 cos k1x 1t 12 cos k2 x 2t •This wave has two kinds of oscillations: •The oscillations at small scales •The “lumps” at large scales 2 cos k2 x 2t Analyzing the sum of two waves: x, t 12 cos k1x 1t 12 cos k2 x 2t cos cos cos sin sin Need to derive some obscure trig identities: •Average these: •Substitute: cos cos cos sin sin 1 2 12 A B 1 2 A B 1 2 cos 12 cos cos cos cos A 12 cos B cos 12 A B cos 12 A B Rewrite wave function: x, t cos kx t cos k x t k1 k2 12 1 2 k 1 2 Small scale oscillations Large scale oscillations k1 k2 12 1 2 k 1 2 The “uncertainty” of two waves 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 k k k2 k 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 cos k x 0 Plotted at t=0 First hint of uncertainty principle k1 k x 12 x k x 1 Group Velocity x, t cos kx t cos k x t Small scale Large scale oscillations 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! vp k vg k More Waves One wave •Two waves allow you to create localized “lumps” Two waves •Three waves allow you to start separating these lumps •More waves lets you get them farther and farther apart Three waves •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 Five waves acts a lot like a particle 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. 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? vp k vg k vp k d vg dk Sample Problem What is the phase and group velocity for this wave? Moved 30 m 30 m vp 1.0 m/s 30 s Finish, t = 30 s Start, t = 0 s Moved 60 m 60 m vg 2.0 m/s 30 s Phase and Group velocity vp How to calculate them: ck •You need the dispersion relation: the relationship between and k, with only constants in the formula v p k k c •Example: light in vacuum has ck What’s wrong with the following proof? Theorem: Group velocity doesn’t always equal phase velocity k d vg dk d d vg ck c dk dk If the dispersion relation is = Ak2, with A a constant, what are the phase and group velocity? kv p d d dv p vg vpk v p k vp dk dk dk Ak 2 vp Ak k k d d 2 vg Ak 2 Ak dk dk 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 •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 k k k x x These quantities are related: •Typically, x k ~ 1 x Precise Relation: (proof hard) xk 12 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 t 12 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/4 L ? L/2 Guess of position L/2 •Best guess: The particle is in the center, x = L/2 •But there is an error x on this amount Exact numbers for x: •It is no greater than L/2 •Particle in a box: 0.181L •It is certainly bigger than 0 •Uniform distribution: 0.289L •Carlson’s rule: use x = L/4 •This rule can be applied in the time domain as well 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: t 12 1 20.0 s 1 2t •Since f = /2, this causes an estimated error of 20.0 s 1 f 3.17 Hz 2 •Of course, the error could be much larger than this Wave Equations You Need: •These equations always apply 1 f T 2 k 2 •Two equations describing a generic wave x, t A cos kx t x, t A sin kx t •Light waves only v p vg c 3108 m/s vp k d vg dk xk 1 2 t 1 2 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 x Re z z x iy of 4 + 7i? y Im z Note: no i z* x iy •Adding, subtracting, and multiplying complex numbers is pretty easy: 2 6 18 i 8 i 24 i 6 10i 24 30 10i 3 4 i 2 6 i •To divide complex numbers, multiply numerator and denominator by the complex conjugate of the denominator 6 8i 18i 24i 2 2 6i 2 6i 3 4i 18 26i 2 9 16i 3 4i 3 4i 3 4i 25 A Useful Identity Taylor series expansion 1 2 1 3 1 4 4 f x f 0 xf 0 2! x f 0 3! x f 0 4! x f 0 sin 3!1 3 5!1 5 7!1 7 9!1 9 cos 1 2!1 2 4!1 4 6!1 6 8!1 8 Apply to sin, cos, and ex functions e 1 x x x x x x In last expression, let x i i e 1 i 1 2! i 2 1 3! i 3 1 4! i 4 1 5! 1 2! 2 i 5 1 3! 1 6! 3 i 1 4! 6 1 7! 4 1 5! i 1 i 2!1 2 3!1 i 3 4!1 4 5!1 i 5 6!1 6 7!1 i 7 1 2!1 2 4!1 4 6!1 6 cos i sin i 3!1 3 5!1 5 7!1 7 ei cos i sin 7 5 x, t A cos kx t Complex Waves Typical waves look like: x, t A sin kx t 1. We’d like to think about them both at once 2. We’d like to make partial derivatives as simple as possible A mathematical trick lets us achieve both goals simultaneously: ik i kx t • Real part is cosine x, t Ae x • Imaginary part is sine This makes the derivatives easier in differential equations: i kx t t i kx t x, t A e ikAe ik x , t x x i kx t i kx t x, t A e i Ae i x , t t t i What is the dispersion relationship for light in vacuum? 2 2 Ez Ez 2 c 0 2 2 t x i 2 Ez c ik Ez 0 2 2 2 c2k 2 Magnitudes of complex numbers The magnitude of a complex number z = x + iy denoted |z|, is given by: •This formula is rarely used 2 2 z x y •The square of the magnitude can be written z x y 2 2 2 x i y x iy x iy zz* z 2 2 This is the easiest way to calculate it 2 2 Sample Problem What’s the magnitude squared of the following expression? 2 2a ix a exp a a it 4a 2 2 2a ix 2a ix a a 2 * exp a exp a a it a it 4a 4a 2 2 2 2a ix 2a ix 8a a exp 2 2 2 a i t 4a 2 2 2i 2 x 2 a x exp exp 2 2 2 2 a t 4a a t 2a a