Problem Set 1 215B-Quantum Mechanics, Winter 2019

Due: Friday, January 18, 2019 by 5pm Put homework in mailbox labelled 215B on 1st ﬂoor of Broida (by elevators).

Problem 1 Sakurai Problems: 3.10, 3.11, 3.12

2: Three-site Antiferromagnet

This problem requires familiarity with angular momentum operators with j > 1/2, as well as addition of angular momentum. So, if needed, please read Sections 3.5 and 3.8 in Sakurai, which should give you the background to do this problem.

Consider a three-site Heisenberg anti-ferromagnet with Hamiltonian

H = J(S~1 · S~2 + S~2 · S~3 + S~3 · S~1), (1) where S~j denote spin operators with spin-S. Here the spin S can be integer or half-odd-integer and the exchange coupling J > 0. a.) How many states are there in the Hilbert space for the three spins? b.) What is the ground state energy, E(S), for arbitrary integer and half-odd-integer spin? [Hint, rewrite the

Hamiltonian in terms of the total spin, S~tot = S~1 + S~2 + S~3.] c.) What is the ground state degeneracy, N(S), for arbitrary integer and half-integer spin S? d.) Treating the spins as classical vectors with magnitude S, evaluate the classical ground state energy Ecl of H and verify that the ratio E(S)/Ecl → 1 as S → ∞.

3: Density Matrices for a spin-1/2 particle ~ (a) Let | ↑i, | ↓i label, as usual, the two basis states of a spin-1/2 particle, Sˆ = (~/2)~σˆ. Consider two different density matrices: (i) A pure state density matrix corresponding to a quantum state with equal amplitude to be spin-up and spin-down, (ii) A density matrix describing a mixture of the two basis states, | ↑i, | ↓i, with equal probability. Show that these two situations lead to the same value for hσˆzi but different values for hσˆxi. (b) Consider the density matrix for a spin-1/2 particle with magnetic moment µ in a magnetic field, with Hamil- tonian, 1 Hˆ = − gµB~ · ~σ,ˆ (2) 2 where g is a constant. Using the equation of motion for the density operatorρ ˆ, find the motion of the polarization vector, P~ = h~σî = T r[ˆρ(t)~σˆ], and compare it with the classical equation of motion of a spinning magnetic dipole in a magnetic field.

4: Block Sphere for Mixed States

A Block sphere representation for a (normalized) pure state of a 2-state system (qubit) is given by:

|ψi = cos(θ/2)|0i + eiφ sin(θ/2)|1i, (3)

Consider a vector of Pauli operators, R~ = (X,ˆ Y,ˆ Zˆ). (a) Show that the expectation value, hψ|R~|ψi, corresponds to a vector on the unit sphere with spherical coordinate angles, (θ, φ). 2

(b) Show that an arbitrary density matrix for a mixed state qubit may be written as,

1ˆ + ~r · R~ ρˆ = . (4) 2 where ~r is a real 3-dimensional vector such that |~r| ≤ 1. This vector is known as the Bloch vector for the stateρ ˆ. (c) Show that a stateρ ˆ is pure if and only if |~r| = 1. (d) What is the Bloch vector for the maximally mixed density matrix with entropy S = −T r(ˆρ logρ ˆ) = log(2)?