Principles of Quantum Mechanics · Problem 8.6.1

The Heisenberg Uncertainty Relations — R. Shankar

Physics — Quantum Mechanics · path-integral-free-particle-propagator

Problem

Verify that

U(x, t; x', 0) = A(t) \exp(iS_{\text{cl}}/\hbar), \quad A(t) = \left(\frac{m}{2\pi\hbar it}\right)^{1/2}

agrees with the exact result, Eq. (5.4.31), for

V(x) = -fx

. Hint: Start with

x_{\text{cl}}(t'') = x_0 + v_0 t'' + \frac{1}{2}(f/m)t''^2

and find the constants

x_0

and

v_0

from the requirement that

x_{\text{cl}}(0) = x'

and

x_{\text{cl}}(t) = x