Quantum theory of electrons in helical magnetic fields

Abstract

We consider the Schr\"odinger equation for a charged spin-\textonehalf{} particle with an intrinsic magnetic moment interacting with a helical magnetic field (a field which is constant and homogeneous at each plane, but which rotates continuously as one is displaced to neighbor parallel planes with some definite frequency). Although the equation defies the exact solution, we take advantage of the facts that a similar equation for a neutral particle can be solved exactly. When the momentum of the charged particle is parallel to the axis of the helix the corresponding exact solution coincides with that of the neutral particle, thereby we develop a perturbative scheme in terms of a transverse velocity-dependent effective coupling constant. Several interesting physical results emerge from these calculations. For example, it is found that the energy spectrum, as a function of the momenta, exhibits a strange band structure having energy gaps only for particular directions in momentum space, contrasting with the neutral-particle spectrum which has no band structure at all. Other quantities like group velocities and the effective mass tensor are computed to lowest order in perturbation theory, giving a nontrivial and curious dependence on the momentum and spin orientation of the particle.