Boundary Conditions in Quantum Mechanics

The quantum mechanical expression for the current density is a function of the external vector potential. As a result, the sources of $\overrightarrow{\mathrm{A}}\left(\overrightarrow{\mathrm{r}}\right)$ can induce spurious sources of $\overrightarrow{\mathrm{j}}\left(\overrightarrow{\mathrm{r}}\right)$, if the wave functions are not carefully chosen. The ideal infinite solenoid is given as an example, and boundary conditions which enforce momentum conservation across the solenoid wall are introduced. As a consequence of these boundary conditions, the wave function in the exterior region is not, in general, single valued; it is a single-valued function multiplied by the well known Dirac phase factor.