Principles of Quantum Mechanics · Problem 21.5

Path Integrals—II — R. Shankar

Physics — Quantum Mechanics · path-integrals-harmonic-oscillator

Problem

Derive the harmonic oscillator propagator

G(x, x'; t) = \sqrt{\frac{m\omega}{2\pi i \hbar \sin(\omega t)}} \exp\left(\frac{im\omega}{2\hbar \sin(\omega t)}\left[(x^2 + x'^2)\cos(\omega t) - 2xx'\right]\right)

by the path integral method. Show all steps including the evaluation of the classical action and the functional determinant.