Franz Schwabl Quantum Mechanics 4th Edit · Problem 7.5

Operators, Matrices, State Vectors

Physics — Quantum Mechanics · plane-rotator-gauge-transformation

Problem

Consider again the plane rotator H0=p022m  ,p0=ℏia∂∂φ  .H_0 = \frac{p_0^2}{2m} \;, \quad p_0 = \frac{\hbar}{\mathrm{i}a}\frac{\partial}{\partial\varphi} \;. Start from the stationary states obeying periodic boundary conditions. Investigate the transformation p0→ps=e−is(φ)/ℏ p0 eis(φ)/ℏ  ,etc.p_0 \to p_s = \mathrm{e}^{-\mathrm{i}s(\varphi)/\hbar}\,p_0\,\mathrm{e}^{\mathrm{i}s(\varphi)/\hbar} \;, \quad \text{etc.} and calculate psp_s, HsH_s, and the new wave functions. Show that the latter in general no longer obey periodic boundary conditions.

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