G25.2651: Statistical Mechanics

Problem set #5

Due: April 7, 2003

1.

The energy of a particle with magnetic moment interacting with a magnetic field is . Consider an electron fixed in space interacting with a magnetic field in the z direction, so that . The electron has a spin of 1/2, and its magnetic moment can be related to its spin by so that the Hamiltonian for the electron becomes

The spin operator where is the spin gyromagnetic ration and , , and are the Pauli matrices given by

a.

Suppose an ensemble of such systems is prepared such that the density matrix initially is

Calculate .

b.

What are the expectation values of the operators , and at any time t?

c.

Suppose now that the initial density matrix is

For this case, calculate .

d.

What are the expectation values of the operators , and at time t for this case?

e.

What are the fluctuations in ? Recall that

f.

Suppose that the density matrix is given initially by a canonical density matrix:

What is ?

g.

What are the expectation values of , and ?

2.

A simple two-state system with energies and is described by a Hamiltonian

where , , and I is the identity operator. Suppose this system has an electric dipole moment and is subject to a laser of frequency with electric field component such that the total Hamiltonian is

Solve the Liouville equation for the density matrix of an ensemble of such systems and calculate the expectation value of in this ensemble.

3.

Consider a one-dimensional free particle in a ``box'' of length L. The Hamiltonian for the system is

a.

Write down a path integral expression for the canonical density matrix as the limit of a P-dimensional integral.

b.

By making the change of variables in the interal,

which is defined recursively in terms of , calculate the density matrix and the partition function for a this system.

c.

Perform the Wick rotation and find an expression for the real-time quantum propagator U(x,x';t).

4.

The partition function for a 2-particle system in one dimension with boson exchange statistics is

Derive a discrete (finite P) path integral expression for the exchange term for the case that

and give an intepretation of your result in terms of a classical mechanical model with points and springs.

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