= Twisted boundary condition
{title2=$\psi(x+L)=e^{i\theta}\psi(x)$}
A twisted <boundary condition> identifies the ends of an interval up to a fixed unit-modulus <complex number>. For the kinetic <Hamiltonian operator> $-\hbar^2\partial_x^2/(2m)$, one also imposes $\psi'(x+L)=e^{i\theta}\psi'(x)$. In <integration by parts>, the endpoint terms then cancel, giving the usual <self-adjoint> ring realization. The allowed <plane waves> have momenta $\hbar(2\pi n+\theta)/L$.
A constant <magnetic vector potential> can be removed from $(p-a)^2/(2m)$ locally by $\widetilde\psi=e^{-iax/\hbar}\psi$. If $\psi$ is periodic, $\widetilde\psi$ has twist $\theta=-aL/\hbar$. Thus the <Aharonov-Bohm effect> survives a locally vanishing vector potential as a global condition on the <wavefunction>.
Back to article page
