 Solution

ID: past-exam-of-the-quantum-condensed-matter-field-theory-course-of-the-university-of-cambridge/2014/paper-1/2/b/solution

The endpoint integral is a Gaussian with coefficient

A = (mω /ℏ) tanh (x /2)

. Therefore

Z = \frac{mω}{2 π ℏ s i n h x} \frac{π}{A} = \frac{1}{2 s i n h x t a n h ( x /2 )} = \frac{1}{2 sinh ( x /2 )} .

It agrees with

\sum_{n \geq 0} e^{- x (n + 1/2)}

, tends to

1/ x

at high temperature, and to

e^{- x /2}

at low temperature. The last limit retains the oscillator zero-point energy.

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