Franz Schwabl Quantum Mechanics 4th Edit · Problem 7.4

Operators, Matrices, State Vectors

Physics — Quantum Mechanics · plane-rotator-aharonov-bohm

Problem

Consider a plane rotator of radius aa described in cylindrical coordinates (r,φ,z)(r,\varphi,z) by the Hamiltonian H0=p022mH_0 = \frac{p_0^2}{2m} with p0=ℏi1a∂∂φ  .p_0 = \frac{\hbar}{\mathrm{i}}\frac{1}{a}\frac{\partial}{\partial\varphi} \;. A plane rotator is a particle of mass mm moving on a circumference of radius aa. (a) Determine the energy eigenfunctions and eigenvalues, imposing periodic boundary conditions. (b) Calculate the wave functions and energy levels for the plane rotator in the presence of a vector potential A=eφ{B2rfor r≤r0(B/2)r02rfor r≥r0  ,\boldsymbol{A} = \boldsymbol{e}_\varphi \begin{cases} \dfrac{B}{2}r & \text{for } r \leq r_0 \\[6pt] \dfrac{(B/2)r_0^2}{r} & \text{for } r \geq r_0 \end{cases}\;, where r0<ar_0 < a. (c) Now consider the vector potential A′=A+∇χ  .\boldsymbol{A}' = \boldsymbol{A} + \boldsymbol{\nabla}\chi \;. For the particular choice χ=−Br022φ  ,\chi = -\frac{Br_0^2}{2}\varphi \;, calculate the vector potential A′\boldsymbol{A}' and the magnetic field B′\boldsymbol{B}'. Pay careful attention to the behavior of B′\boldsymbol{B}' at the origin. Does the transformation A→A′\boldsymbol{A}\to\boldsymbol{A}' represent a gauge transformation? (d) Find the wave functions and the energy eigenvalues for A\boldsymbol{A}. (e) Compare the results of (b) and (d) and explain why the Aharonov--Bohm effect is not contradicted by the transformation from A\boldsymbol{A} to A′\boldsymbol{A}'.

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