Electron bound by a potential well in the presence of a constant uniform magnetic field

Abstract

We study the effect of a constant uniform magnetic field on an electrically charged massive particle (an electron) bound by a potential well, which is described by means of a single attractive $λδ({\bf r})$ potential. A transcendental equation that determines the electron energy spectrum is derived and solved. The electron wave function in the ground (bound) state is approximately constructed in a remarkable simple form. It is shown that there arises the probability current in the bound state in the presence of a uniform constant magnetic field. This (electric) current, being by the gauge invariant quantity, must be observable and involve (and exercise influence on) the electron scattering. The probability current density resembles a stack of "pancake" vortices'' whose circulating "currents'' around the magnetic field direction ($z$-axes) are mostly confined within the plane $z=0$. We also compute the tunnelling probability of electron from the bound to free state under a weak constant homogeneous electric field, which is parallel to the magnetic field. The model under consideration is briefly discussed in two spatial dimensions.