Quantum Mechanics C (130C) Winter 2014 Assignment 7

Transcription

1 University of California at San Diego Department of Physics Prof. John McGreevy Quantum Mechanics C (130C) Winter 014 Assignment 7 Posted March 3, 014 Due 11am Thursday March 13, 014 This is the last problem set. Problem Set 7 1. Stationary phase. [extra credit] (a) What is N n=1 einθ? Consider both the case when θ is an integer multiple of π/n, and when it is not. (b) Add together some random phases, i.e. N i=1 eiϕ i where {ϕ i } are chosen randomly and uniformly in the interval [0, π). (The Mathematica command RandomReal is useful here.) How does the magnitude of the answer depend on N as N grows? (c) Evaluate numerically the integral I(N) = inv (x) dxe with V (x) = (x 1) 1 x3 for several values of N, and compare to the answer from the stationary-phase approximation. [The range of integration shouldn t matter too much, but you can take x (, ). If you are worried about convergence of the integral, you can define it by I(N) = for arbitrarily small positive epsilon.]. Path integral for a free particle dxe π ei( ɛ) NV (x) Consider the path integral description of the quantum mechanics of a free particle in one dimension. The action is S[x] = tf (a) What is the equation of motion 0 = δs δx(t)? 0 dt m ẋ. (b) Find the classical solution x(t) with x(t = 0) = x 0 and x(t = t f ) = x f. 1

2 (c) Evaluate the action for the classical solution S[x], and evaluate the stationaryphase approximation to the path integral for the quantum propagator U(x f, t f ; x 0, 0) = x f U(t f ) x 0 = [dx]e is[x] U sc e is[x]. (d) Derive the Hamiltonian associated to the action S = dtl above. [That is, find p = L and eliminate ẋ in H(x, p) = pẋ L.] ẋ (e) Treating this Hamiltonian quantum mechanically, evaluate the exact quantum propagator, U(x f, t f ; x 0, 0) = x f U(t f ) x 0 = x f e it f H x 0. [One way to do this is to use the fact that U here solves the ODE i t U(x, t; x 0, 0) = m xu(x, t; x 0, 0) with the initial condition U(x f, 0; x 0, 0) = δ(x f x).] Compare with the semiclassical approximation U sc defined above. 3. A charged particle, classically. [Extra credit] This problem is an exercise in calculus of variations, as well as preparation for our discussion of particles in electromagnetic fields. Consider the following action functional for a particle in three dimensions: ( m S[x] = dt x eφ( x) + e c x A(x)). (a) Show that the extremization of this functional gives the equation of motion: δs[x] δx i (t) = mẍi (t) e x iφ(x(t)) + e cẋj F ij (x(t)) where F ij x ia j x ja i. Show that this is the same as the usual Coulomb- Lorentz force law ( F = e E + v ) c B with B i ɛ ijk F jk. (b) Show that the canonical momenta are Π i L ẋ i = mẋi + e c A i(x). Here S = dtl. (I call them Π rather than p to emphasize the difference from the mechanical momentum mẋ.) Show that the resulting Hamiltonian is H ẋ i Π i L = 1 ( Π i e i(x(t))) m c A + eφ. i

3 4. Coherent states. Consider a quantum harmonic oscillator. The creation and annihilation operators a and a satisfy the algebra [a, a ] = 1 and the vacuum state 0 satisfies a 0 = 0. Coherent states are eigenstates of the annihilation operator: (a) Show that a α = α α. α = e α / e αa 0 = e α / α n n! n is an eigenstate of a with eigenvalue α. (a is not hermitian, so its eigenvalues need not be real.) (b) Coherent states with different α are not orthogonal. (a is not hermitian, so its eigenstates need not be orthogonal.) Show that α 1 α = e α 1 α. (c) Compute the expectation value of the number operator n = a a in the coherent state α. (d) Time evolution acts nicely on coherent states. The hamiltonian is H = ω ( a a + 1 ). Show that a coherent state evolves into a coherent state with an eigenvalue α(t): where α(t) = e iωt α. e iht α = e iωt/ α(t) 5. Two coupled spins. [based on Le Bellac problem 6.5.4] Consider a four-state system consisting of two qbits, H = span{ ɛ 1 ɛ ɛ 1 ɛ, ɛ = z, z }. (a) For each qbit, define σ ± 1 (σx ± iσ y ). (These are raising and lowering operators for σ z : [σ z, σ ± ] = ±σ ±. Show this.) Show that σ 1 σ = ( ) σ 1 + σ + σ1 σ + + σ z 1 σ. z (b) Determine the action of the operator σ 1 σ on the basis states (c) Show that the four vectors,,,. 0, 0 = 1 ( ), 1, 1, 1, 0 1 ( + ), 1, 1 are orthonormal and are eigenvectors of σ 1 σ with eigenvalues 1 or 3. 3

4 (d) Show that they are also eigenvectors of J ( σ 1 + σ ) and J z σ z 1 + σ z and find their eigenvalues. (e) Consider the operator P 1, 1 (1 + σ 1 σ ) acting on the two spins. Show that P 1, acts by exchanging the states of the two spins: P 1, ɛ 1 ɛ = ɛ ɛ Spin chains and spin waves. [Related to Le Bellac problem on page 00] A one-dimensional ferromagnet can be represented as a chain of N qbits (spin-1/ particles) numbered n = 0,...N 1, N 1, fixed along a line with a spacing l between each successive pair. It is convenient to use periodic boundary conditions (as in HW problem ), where the Nth spin is identified with the 0th spin: n + N n. Suppose that each spin interacts only with its two nearest neighbors, so the Hamiltonian can be written as H = 1 NJ1 1 N 1 J σ n σ n+1. where J is a coupling constant determining the strength of the interactions. (a) Show that all eigenvalues E of H are non-negative, and that the minimum energy E 0 (the ground state) is obtained in the state where all the spins point in the same direction. A possible choice for the ground state Φ 0 is then Φ 0 = z z n=1... z N (b) Show that any state obtained from Φ 0 by rotating each of the spins by the same angle is also a possible ground state. [Hint: the generator of spin rotations J n σ n commutes with the Hamiltonian.] [Cultural remark: the phenomenon of a ground state which does not preserve a symmetry of the Hamiltonian is called spontaneous symmetry breaking. ] (c) Now we wish to find the low-energy excitations above the ground state Φ 0. Show that H can be written where H = NJ1 J N 1 P n,n+1 = J N 1 P n,n+1 1 (1 + σ n σ n+1 ). (1 P n,n+1 ). Using the result of the previous problem, show that the eigenvectors of H are linear combinations of vectors in which the number of up spins minus the number of down spins is fixed. Let Ψ n be the state in which the spin n is down with all the other spins up. What is the action of H on Ψ n? 4

5 (d) We are going to construct eigenvectors k s of H out of linear combinations of the Ψ n. Let k s = N 1 e iksnl Ψ n with k s = πs, s = 0, 1,...N 1. Nl Show that k s is an eigenvector of H and determine the energy eigenvalue E k. Show that the energy is proportional to ks as k s 0. This state describes an elementary excitation called a spin wave or magnon with wave-vector k s. 7. A damped quantum harmonic oscillator [extra credit] [from Preskill] In this problem we extend the amplitude damping channel which we described for a qbit to a harmonic oscillator. This arises if we couple a harmonic oscillator with creation operator a to an environment via an interaction like H = i λ i ab i + h.c. (1) Here we have modeled the environment by a collection of oscillators whose modes are created by b i. For the first part of the problem, let s instead model the environment by two-state system. Consider a situation where evolution over a time dt is given by U n 0 E = 1 p n 0 E + pa n 1 E. In this expression we suppose that p 1 and we will work only to leading order in p. (a) Note that a n n 1. Find the proportionality constant. (b) Find the Kraus operators M 0, M 1 for the evolution above and show that they satisfy the unitary condition 1 = M 0 M 0 + M 1 M 1 to leading order in p. (c) Find the evolution of the density matrix. To do better than the above, let s introduce a little more technology. Suppose we are studying an environment which forgets very rapidly. Then we may take the limit where dt 0 and write a differential equation for the density operator. For sure we have ρ(dt) = M µ ρ(0)m µ = ρ(0) + O(dt) and so one of the Kraus operators is M 0 = 1 + O(dt), and the others are small, of order dt, describing transitions that may occur with probability proportional to dt. Let s make the ansatz M i = dtl i, i = 1,, 3 M 0 = 1 + dt ( ih + K) 5

6 where H is the (hermitian) Hamiltonian for the subsystem (H = ω ( a a + 1 ) for the SHO) this term describes the unitary evolution that would happen if the system were closed and K is some new Hermitian operator describing the damping. Each of H, L i, K is zeroth order in dt. (d) Determine K in terms of the L i using the unitarity condition. You should find that t ρ = L[ρ] i[h, ρ] + ( L i ρl i 1 L i L iρ 1 ) ρl i L i. () i The superoperator on the RHS is called the Lindbladian. (e) Now we return to the situation above in (1). In this case there is one lindblad operator, L 1 = Γa. (Think of Γ as the rate for one of the environment oscillators b to decay from the first excited state to the groundstate.) Apply the evolution in () to the density matrix, and show that the occupation number of the oscillator n a a satisfies t n(t) = tra a ρ = Γ n(t). Integrate this equation and show that the oscillator is damped. (f) [Note: It is possible to do this part starting from eqn (), even if you did not do the previous parts.] Now we will think about decoherence. The fact that the oscillator is coupled to the environment via the operator a suggests that it is eigenstates of a that will be the classical states. This turns out to be correct. As we saw in problem 4, eigenstates of a are coherent states. Suppose we begin in a Schrödinger s cat state of the form 1 ( α 1 + α ), with α 1 α 1. The initial density matrix is ρ(0) = 1 ( α 1 + α ) ( α 1 + α ). Show that, as a result of the coupling to the environment above, the off-diagonal terms decay rapidly: ρ(t) = 1 ( α 1 α 1 + α α ) + 1 e 1 Γt α 1 α ( α α 1 + h.c.). So the decoherence rate is Γ dec = 1 α 1 α Γ, much faster than the damping rate, if the coherent states are very different. [First hint: It may be useful to write ρ(0) = β i β j e α ia 0 0 e α j a ij=1, with β i = 1 e 1 α i, and to make the ansatz ρ(t) = β ij (t)e α ia 0 0 e α j a. ij=1, 6

7 The claim is that for tγ 1, β ii (t) α i, but ] β 1 (t) e 1 α 1 α Γt α 1 α. 7