Principles of Quantum Mechanics · Problem 21.8

Path Integrals—II — R. Shankar

Physics — Quantum Mechanics · path-integrals-statistical-mechanics

Problem

The partition function in statistical mechanics is

Z = \text{Tr}\left(e^{-\beta H}\right) = \int dx\, \langle x | e^{-\beta H} | x \rangle.

Show that this can be written as a path integral over periodic paths

x(0) = x(\beta\hbar)

in imaginary time

\tau = it

Z = \oint \exp\left(-\frac{1}{\hbar} \int_0^{\beta\hbar} \left[\frac{m}{2}\dot{x}^2 + V(x)\right] d\tau\right) \mathcal{D}[x(\tau)].