PHY 6645 Fall 2002 – Homework 10
Due by noon on Friday, December 13. No credit will be available for homework
submitted after this deadline. Solutions to this assignment will be available for collection
outside Kevin Ingersent’s office (NPB 2612) beginning immediately after the deadline.
Answer all questions. To gain maximum credit you should explain your reasoning and show
all working. Please write neatly and include your name on the front page of your answers.
1. Shankar Exercise 14.3.4.
2. Shankar Exercise 14.3.6.
3. Shankar Exercise 14.3.8, modified so that the words “matrix” and “matrices” are replaced everywhere by “operator” and “operators” (with the understanding that these
are operators acting on a two-dimensional linear vector space). In other words, work
with the abstract representation of the Pauli operators, rather than a concrete representation of the Pauli matrices (such as that in the basis of eigenstates of σz ).
4. A beam of electrons is passed through a region containing a spatially uniform, temporally constant magnetic field B = B n̂, where n̂ is a unit vector. Each electron is
described by a Hamiltonian H = Ho + Hs . Ho acts purely on the orbital (spatial)
degrees of freedom, i.e., it is a function of the position and momentum operators.
Hs = −γB · S (where γ is a positive, real constant) acts purely on the spin degrees of
freedom. You may assume, therefore, that [Ho , Hs ] = 0.
Suppose that the electrons are in thermal equilibrium at temperature T . Then quantum
statistical mechanics tells us that their state operator is
ρ = A exp(−H/kB T ),
where kB is Boltzmann’s constant and A is a normalization factor.
(a) Show that the density operator for this system factorizes into orbital and spin
parts: ρ = ρo ⊗ ρs .
For the remainder of the problem, we will ignore ρo , and focus on ρs , which acts on
the two-dimensional vector space describing a single spin- 12 degree of freedom.
(b) Prove that the most general spin-state operator has the form ρs = 12 (I + p · σ),
where σj = 2Sj /h̄ (j = 1, 2, 3) is a Pauli operator and p is a real “polarization
vector,” whose length cannot exceed 1.
Hint: You may carry out your proof using the abstract operators, or if you prefer,
using a matrix representation of the operators.
(c) Calculate the polarization vector p for the electron beam in a magnetic field B
at temperature T .
(d) Find the eigenvalues of the state operator for the electron beam in a magnetic
field B at temperature T . Show that the eigenkets are the kets |n̂up i and |n̂down i
discussed on Shankar pp. 379 and 380.
(e) Calculate the expectation value hn̂ · Si.
(f) Calculate the expectation values of the three Cartesian components of the spin:
hSj i, j = 1, 2, 3. Hence, find a relation between hSi and p.