Principles of Quantum Mechanics · Problem 21.7

Path Integrals—II — R. Shankar

Physics — Quantum Mechanics · path-integrals-perturbation-theory

Problem

Using the path integral formulation, derive the first-order perturbation theory correction to the propagator for a potential

V = V_0 + V_1

, where

V_1

is small. Show that

G(x, x'; t) \approx G_0(x, x'; t) - \frac{i}{\hbar} \int_0^t dt'' \int dx'' G_0(x, x''; t - t'') V_1(x'') G_0(x'', x'; t'').