 Solution

Define a transformed wavefunction byIndeed, writing $\psi =e^{iax/\hslash }\tilde{\psi }$ givesThis is the electromagnetic gauge transformation generated by $\chi (x)=-A_{x}x$: $A_x^{\prime }=A_x+{\partial }_x\chi =0$ and ${\psi }^{\prime }=e^{iq\chi /\hslash }\psi$. The transformed Hamiltonian operator is consequently $p^2/(2m)$, but the original periodic boundary condition becomes the twisted boundary conditionFor the domain of the self-adjoint kinetic Hamiltonian operator, the derivative has the same twist, ${\tilde{\psi }}^{\prime }(x+L)=e^{-iaL/\hslash }{\tilde{\psi }}^{\prime }(x)$. Thus the magnetic flux has moved into the boundary condition. The transformation is a single-valued gauge on periodic representatives only when $aL/\hslash$ is an integer multiple of $2\pi$; otherwise it is understood on the cut interval with the displayed identification of its endpoints.