## Midterm II reflection, take II, with approximate anharmonic oscillator solution

Posted by peeterjoot on March 11, 2013

## Question: Perturbation of classical harmonic oscillator (2013 midterm II p2)

Consider a single particle perturbation of a classical simple harmonic oscillator Hamiltonian

Calculate the canonical partition function, mean energy and specific heat of this system.

This problem can be attempted in two ways, the first of which was how I did it on the midterm, differentiating under the integral sign, leaving the integrals in exact form, but not evaluated explicitly in any way.

That solution was posted previously.

Alternately, by Taylor expanding around and with those as the variables in the Taylor expansion (as now done in the Pathria 3.29 problem), we can form a solution in short order. Given my low midterm mark, it seems very likely that this was what was expected.

Performing a two variable Taylor expansion of , about we have

or

Now we can calculate the average energy

Dropping the , terms of the denominator above, we have

The heat capacity follows immediately

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