Principles of Quantum Mechanics · Problem 21.4

Path Integrals—II — R. Shankar

Physics — Quantum Mechanics · path-integrals

Problem

Derive the propagator for a particle in a uniform gravitational field, V=mgxV = mgx. That is, find G(x,x′;t)=∫x(0)=x′x(t)=xexp⁡(iℏ∫0t[12mx˙2−mgx]dt′)D[x(t′)].G(x, x'; t) = \int_{x(0)=x'}^{x(t)=x} \exp\left(\frac{i}{\hbar} \int_0^t \left[\frac{1}{2}m\dot{x}^2 - mgx\right]dt'\right) \mathcal{D}[x(t')]. Hint: The classical path is xcl(t′)=x′+(x−x′)tt′−12gt′2+12gt′tx_{cl}(t') = x' + \frac{(x - x')}{t}t' - \frac{1}{2}g t'^2 + \frac{1}{2}g t' t, and the fluctuations around it satisfy a free-particle equation.

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