Principles of Quantum Mechanics · Problem 21.12

Path Integrals—II — R. Shankar

Physics — Quantum Mechanics · path-integrals-gaussian

Problem

Evaluate the Gaussian path integral

I = \int_{x(0)=0}^{x(T)=0} \exp\left(\frac{i}{\hbar} \int_0^T \frac{m}{2}\dot{x}^2\, dt\right) \mathcal{D}[x(t)]

by expanding

x(t)

in a Fourier sine series:

x(t) = \sum_{n=1}^{\infty} a_n \sin\left(\frac{n\pi t}{T}\right).

Show that the result is consistent with the free-particle propagator evaluated at

x = x' = 0