G25.2651: Statistical Mechanics

Problem set #6

Due April 14, 2003

1.

Consider two distinguishable particles in one dimension with respective coordinates x and y and conjugate momenta and with a Hamiltonian

a.

Show that the density matrix can be written in the form

where T[x;y,y'] is known as the influence functional. What is the functional integral expression for T[x;y,y'], and of what function is T[x;y,y'] a functional?

b.

Give a closed form expression for by evaluating the functional integral.

: Use the method of expansion about the classical path, and treat as an arbitrary function of time, i.e. as a forcing function in the equation of motion for .

2.

a.

Show that the path integral expression for the density matrix can be written as:

b.

Consider a double well potential of the form

Show that, for a particle of unit mass, the dominant path for the density matrix is given by

in the low temperature limit with negligible error in the endpoint conditions. This path is called an instanton or kink solution. Discuss the behavior of this trajectory in imaginary time .

c.

Calculate the classical imaginary-time action for the kink solution.

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