1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 405

After all, one still needs to solve the eigenvalue problem for the Hamiltonian, which
involves solution eqn. (10.1.5). In Section 10.1, we described the difficulty inherent
in this approach. The eigenvalue equation can be solved explicitly only for systems
with a very small number of degrees of freedom. Looking ahead, in Chapter 12 we will
develop a framework, known as the Feynman path integral formulation of statistical
mechanics, that allows the calculation ofN-particle eigenvalues to be circumvented,
thereby allowing quantum equilibrium properties of large condensed-phase systems to
be evaluated using molecular dynamics and Monte Carlo methods. Before exploring
this formalism, however, we will use the traditional eigenvalue approach to study the
quantum ideal gases, the subject of the next chapter.

10.5 Problems

10.1. a. Prove that the trace of a matrix A is independent of the basis in which

the trace is performed.

b. Prove the cyclic property of the trace

Tr(ABC) = Tr(CAB) = Tr(BCA).

10.2. Recall from Problem 9.1 that the energy of a quantum particle with magnetic

momentμin a magnetic fieldBisE=−μ·B. Consider a spin-1/2 particle

such as an electron fixed in space in a uniform magnetic field in thezdirection,

so thatB= (0, 0 ,B). The Hamiltonian for the particle is given by

Hˆ=−γBSˆz.

The spin operators are given in eqn. (9.4.2).

a. Suppose an ensemble of such systems is prepared such that thedensity

matrix initially is

ρ ̃(0) =

(

1 /2 0

0 1/ 2

)

.

Calculate ̃ρ(t).

b. What are the expectation values of the operatorsSˆx,Sˆy, andSˆzat any

timet?

c. Suppose now that the initial density matrix is

̃ρ(0) =

(

1 / 2 −i/ 2

i/2 1/ 2

)

.

For this case, calculate ̃ρ(t).