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International Journal of Modern Physics A

On an electron in a nonuniform axial magnetic field in a uniformly rotating frame

K. Bakke, R. F. Ribeiro, C. Salvador

Published Nov 30, 2019• 3 citations
Publisher

Abstract

The interaction of an electron with a nonuniform axial magnetic field is analyzed in a uniformly rotating frame. In particular, the magnetic field is proportional to the square of the radial distance from the symmetry axis. Then, in search of analytical solutions to the Schrödinger equation, it is shown that these solutions are possible if the nonuniform magnetic field possesses a discrete set of values.

DOI
10.1142/s0217751x19502294

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The interaction of an electron with a nonuniform axial magnetic field is analyzed in a uniformly rotating frame. In particular, the magnetic field is proportional to the square of the radial distance from the symmetry axis. Then, in search of analytical solutions to the Schrödinger equation, it is shown that these solutions are possible if the nonuniform magnetic field possesses a discrete set of values.
From the interaction between the magnetic quadrupole moment of a neutral particle (atom or molecule) with a radial electric field, a uniform effective magnetic field perpendicular to the plane of motion of the neutral particle appears. In addition, it is considered that the presence of a scalar potential is proportional to the radial distance and noninertial effects that stem from a rotating reference frame. Then, analytical solutions to the Schrödinger equation are obtained, where it is shown that the Landau-type levels are modified, and the allowed values of the cyclotron frequency are determined by the angular velocity of the rotating frame, the parameter associated with the scalar potential proportional to the radial distance and the quantum numbers of the system.
The behaviour of the interaction of the induced electric dipole moment of an atom with a uniform magnetic field and a non-uniform electric field are investigated in a rotating reference frame. An interesting aspect of this interaction is that it gives rise to an analogue of a spinless particle subject to the doubly anharmonic oscillator. Then, it is shown that analytical solutions to the Schrödinger equation can be obtained. Another point raised is that the quantum effects on the induced electric dipole moment can be observed if the uniform magnetic field possesses a discrete set of values.
We study some aspects of the quantum theory of a charged particle moving in a time-independent, unidirectional magnetic field. When the field is uniform, we make a few clarifying remarks on the use of angular momentum eigenstates and momentum eigenstates with the diamagnetism of a free electron gas as an example. When the field is nonuniform but weakly varying, we discuss both perturbative and nonperturbative methods for studying a quantum mechanical system. As an application, we derive the quantized energy levels of a charged particle in a Helmholtz coil, which go over to the usual Landau levels in the limit of a uniform field.
The behavior of an atom in a nonuniform magnetic field is analyzed, as well as the motion of a classical magnetic dipole (a spinning charged ball) and a rotating charged ring. For the atom it is shown that, while the magnetic field does no work on the electron-orbital contribution to the magnetic moment (the source of translational kinetic energy being the internal energy of the atom), whether or not it does work on the electron-spin contribution to the magnetic moment depends on whether the electron has an intrinsic rotational kinetic energy associated with its spin. A rotational kinetic energy for the electron is shown to be consistent with the Dirac equation. If the electron does have a rotational kinetic energy, the acceleration of a silver atom in a Stern-Gerlach experiment or the emission of a photon from an electron spin flip can be explained without requiring the magnetic field to do work. For a constant magnetic field gradient along the z axis, it is found that the classical objects oscillate in simple harmonic motion along the z axis, the total kinetic energy -- translational plus rotational -- being a constant of the motion. For the charged ball, the change in rotational kinetic energy is associated only with a change in the precession frequency, the rotation rate about the figure axis remaining constant.