Physics 6311: Statistical Mechanics - Homework 6

due date: Tuesday, Oct 7, 2025

_________________________________________________________________________________________

Problem 1: Generalized equipartition theorem (10 points)

Consider a classical Hamiltonian of the form

H = \sum_{i = 1}^{3 N} \frac{1}{2} A_{i} | q_{i} |^{n} + ∑_{i = 1}^{3 N} \frac{1}{2} B_{i} p_{i}^{2}

where $n > 0$ is an exponent that characterizes the potential energy and $A_{i}$ and $B_{i}$ are positive constants. Using the canonical ensemble, calculate the internal energy and the specific heat at constant volume as functions of temperature.

Problem 2: Quantum harmonic oscillator (15 points)

Consider a quantum harmonic oscillator given by the Hamilton operator

Ĥ = \frac{{\hat{p}}^{2}}{2 m} + \frac{m}{2} ω^{2} {\hat{x}}^{2}

where $m$ is the mass and $ω$ is the oscillator frequency. The oscillator is in contact with a heat bath at temperature $T$ .

a): Write down the eigenstates of the Hamiltonian and their energies.
b): Using the canonical ensemble, compute the partition function of the oscillator.
c): Find the average energy and the specific heat as functions of temperature. Compare with the results of the classical oscillator discussed in class.
d): Find the entropy and show that it fulfills the third law of thermodynamics.

Problem 3: Pendulum (15 points)

Consider a (classical) pendulum consisting of a point mass $m$ attached to the pivot point via a massless rod of length $L$ . The pendulum is coupled to a heat bath at temperature $T$ , and its motion is restricted to a vertical plane.

a): Using the canonical ensemble, write down the partition function. Show that it factorizes into kinetic and potential parts.
b): Find the average angular velocity of the pendulum and its standard deviation as functions of temperature.
c): Evaluate the partition function in the limit of small vibrations. Find the average energy and the specific heat of the pendulum in this limit.
d): Calculate the lowest order in $T$ corrections to the small-vibration results for the average energy and the specific heat. (To do so, expand the potential energy about its minimum).