PHYS 508 (2015-1) Final Exam January 27, Wednesday. Q1. Scattering with identical particles The quantum statistics have some interesting consequences for the scattering of identical particles. This is best seen for scatterers with only a few final states. As an example, consider a partially-silvered mirror as a scatterer for particles. Light incident from channel a will scatter along p and q directions. Similarly, light incident from b channel will scatter into the same two directions, p and q. We can investigate this scattering process classically for classical electromagnetic waves as well as quantum mechanically for photons. This type of scattering is not specific to photons only. It is possible to create small semiconductor structures where electrons are scattered just like that (i.e., electrons can be incident from two possible incoming channels and they scatter into two possible outgoing channels). Hence, we will assume that some type of particles are scattering from such a mirror and investigate the process quantum mechanically. The particles can be of any type, bosons or fermions; it will not matter. We will still use the partially-silvered mirror picture however, because it is simple. Below, we will first investigate the scattering of a single particle and its quantum mechanical representation. After that, we will look at the scattering of two or more particles. Scattering of a single particle: Suppose that a particle is incident from channel a. We can show the incident state by a in ket representation. If you want, it is possible to use the wavefunction φ a ( r) to show this state. After passing the mirror, the particle s wavefunction will be split into two coherent superpositions of p and q 1. We can think of this as follows: Let τ be a time during which the particle crosses the mirror. Let U(τ) be 1 Coherence will not be lost unless a detector detects the particle and determines the channel it has taken. If there is no measurement, however, the coherence will not be lost.
Page 2 of 5 the corresponding time development operator. We say that the incident state a evolves within time τ to a U(τ) a = A p + B q, (1) or φ a ( r) U(τ)φ a ( r) = Aφ p ( r) + Bφ q ( r). We can do the same for a particle incident from channel b. In that case too, the particle will be scattered into a superposition of p and q. Suppose that b U(τ) b = C p + D q. (2) You can easily rewrite the same equation in terms of wavefunctions. It is also possible to consider the incident particle to be in a superposition state like c 1 a + c 2 b. This can happen for Mach-Zehnder interferometers where there are two partially-silvered mirrors. The first mirror is used for splitting the wavefunction into two. The two channels are then guided to the input channels of the second mirror. (a) For the incident state ψ in = c 1 a +c 2 b, find the outgoing state ψ out = U(τ) ψ in. Then, using the fact that U(τ) is unitary and hence norm-preserving (i.e., ψ out 2 = ψ in 2 ), show that the scattering matrix [ ] A C S = B D is unitary. (a ) Using S S = I and SS = I, verify that each column and each row of S are normalized vectors. (b) Consider an experiment where detectors measure the final channel the particle has taken. Let T (a q) be the probability that when the particle is incident from channel a, it will be detected in channel q (this is usually named as the transmission probability ). Using unitarity of the matrix S, show that T (a q) = T (b p), in other words, the transmission probability is the same for both possible incidence directions. For any such mirror, we can re-define the overall phases of the states b, p and q in such a way that all scattering amplitudes (matrix elements of S) are real numbers. Hence, for some parameter α, the scattering by the mirror can be expressed as For questions below, use these values. a U(τ) a = cos α p + sin α q, b U(τ) b = sin α p + cos α q,
Page 3 of 5 (c) Compute the transmission T = T (a q) = T (b p) and the reflection R = T (a p) = T (b q) probabilities. Scattering of two incident particles: Let us now look at our real problem. Suppose that two particles are sent to the mirror in such a way that one comes from a and the other comes from b. We also place detectors at the end of arms p and q and count the number of particles scattered into each arm. An important question is how we can represent the states. Obviously this is a two-particle state and hence the wavefunction depends on the coordinates of the two particles. Let us just consider the case where the particles are distinguishable. The incident state is then ψ in ( r 1, r 2 ) = φ a ( r 1 )φ b ( r 2 ) We can represent the same state in the Dirac bra-ket notation as well ψ in 12 = a 1 b 2 where i reminds us that the ket is used for the ith particle and the symbol is just a reminder that we are taking products of wavefunctions of two independent particles. Of course, if the particles are identical, properly (anti)symmetrized ψ in has to be constructed. Ignore the spin/polarization of the particles. For example, suppose that all particles have the same spin/polarization state and the scattering by mirror does not change these. (d) Consider the initial state where one particle is coming from a and the other is coming from b. Write the correct initial initial state ψ in that must be used for the following cases. After that, compute the final state after the scattering for each case. ψ out 12 = U 1 (τ) U 2 (τ) ψ in i. When the particles are distinguishable. ii. When the particles are identical bosons. iii. When the particles are identical fermions. (d ) For each case above, find the probability of each possible outcome of the detectors. Present your results as a table for easy comparison. Discuss the differences between them. i. Distinguishable ii. Boson iii. Fermion D p detects 2, D q detects 0......... D p detects 1, D q detects 1......... D p detects 0, D q detects 2......... Total Probability.........
Page 4 of 5 (d ) Consider the special case of semi-silvered mirror for which α = π/4. In that case, the reflection and transmission are equally probable, T = R = 1/2. Compute the probabilities for this special case, rewrite the table and compare. Q2. Average velocity of a Dirac particle Consider the free Dirac equation. We have said that the velocity operator is given by v = c α. In this problem, we will check if the velocity-momentum relation is satisfied as expected. (a) First, let us write the relations classically. For a particle with mass m, (i) express p in terms of v, and (ii) express v in terms of p. Please note: Both relations should show the explicit (and not implicit) dependence of one to the other. In other words, by inserting the value of one to the appropriate equation, you should be able to get the other. (b) Consider the momentum eigenstates of the free Dirac equation. For a given momentum value, two of these eigenstates correspond to the positive energy and the other two correspond to the negative energy. As we will compute an expectation value, we should normalize these. Assume that the particle is inside a finite (but large) volume V, and use the periodic-boundary conditions ideas to write down all four of the momentum eigenstates for a given value of momentum. (c) Compute v i for all momentum eigenstates and compare with the formula given in part (a). (Especially discuss the relative directions of velocity and momentum for positive and negative energy solutions.) (d) Find the 4-current density J µ for all momentum eigenstates and interpret. Q3. Spin of a Dirac particle We know that the spin operator S = ( /2) Σ does not commute with the Dirac Hamiltonian H D. This implies that the spin is not conserved. It also implies that there is no state where both are certain. As a result, for the energy eigenstates you have used in Q2, there should be an uncertainty in spin. In this problem, we are going to compute this uncertainty. Let ˆn = sin θ(cos φˆx + sin φŷ) + cos θẑ be a direction in space. The 2 1 spinor we use to represent the spin-up state along ˆn in non-relativistic quantum mechanics is [ ] cos θ φˆn = 2 sin θ. 2 eiφ We will insert φˆn into the large component of the Dirac spinor ψ. (a) First of all, compute explicitly and verify that φ φ = 1 and φ σφ = ˆn for φ = φˆn.
Page 5 of 5 (b) Find ( σ p)σ i ( σ p) by using the Pauli spin matrix algebra: σ i σ j = δ ij I + iɛ ijk σ k, or ( σ a)( σ b) = ( a b)i + i σ ( a b). Also you may need: ɛ ijk ɛ lmk = δ il δ jm δ im δ jl, a ( b c) = b( a c) c( a b). (b ) Compute φ ( σ p)σ i ( σ p)φ for φ = φˆn. (c) Consider the momentum eigenstates of the Dirac equation with positive energy. Choose the large component to be φ = φˆn. Find the corresponding small component χ. After that compute the average spin in the momentum eigenstate Σ = d 3 r ψ Σψ. (You may want to use part (b)-(b ) in here.) (c ) Let us use the unit vector â to show the direction of momentum, p = pâ. Simplify the expression for Σ as much as possible by expressing it in terms of ˆn, â and E (i.e., eliminate the magnitude of momentum from your expression). (d) Now, we can look at a few special cases. V i. First look at the case where momentum p is parallel or anti-parallel to ˆn, i.e., â = ±ˆn. What is Σ for this case? ii. Now, look at the case where momentum p is perpendicular to ˆn, i.e., â ˆn. What is Σ for this case? (e) Let Σ n = Σ ˆn be the component of spin along ˆn. What is Σ 2 n? What is the uncertainty Σ n? (f) Compute the uncertainty for the case (i.) and (ii.) of part (d) and interpret the results.