Principles of Quantum Mechanics · Problem 21.19

Path Integrals—II — R. Shankar

Physics — Quantum Mechanics · path-integrals-topology

Problem

A particle moves on a ring of radius

R

(i.e.,

x \in [0, 2\pi R)

with periodic boundary conditions). (a) Show that the propagator receives contributions from all topological sectors labeled by the winding number

n

G(\theta, \theta'; t) = \sum_{n=-\infty}^{\infty} G_n(\theta + 2\pi n, \theta'; t).

(b) If there is an Aharonov-Bohm flux

\Phi

through the ring, show that each winding number sector acquires a phase

e^{in\Phi e/\hbar c}