Pauli's Electron in Ehrenfest and Bohm Theories, a Comparative Study

Abstract

Electrons moving at slow speeds much lower than the speed of light are described by a wave function which is a solution of Pauli's equation. This is a low-velocity limit of the relativistic Dirac equation. Here we compare two approaches, one of which is the more conservative Copenhagen's interpretation denying a trajectory of the electron but allowing a trajectory to the electron expectation value through the Ehrenfest theorem. The said expectation value is of course calculated using a solution of Pauli's equation. A less orthodox approach is championed by Bohm, and attributes a velocity field to the electron also derived from the Pauli wave function. It is thus interesting to compare the trajectory followed by the electron according to Bohm and its expectation value according to Ehrenfest. Both similarities and differences will be considered.

Keywords: Ehrenfest theorem; quantum mechanics; spin.

Figure 1

A schematics of Stern–Gerlach experiment. Neutral particles travelling through an inhomogeneous magnetic field, and being deflected up or down depending on their spin; (1) particle source, (2) beam of particles, (3) inhomogeneous magnetic field, (4) classically expected result (neglecting the quantum spin force), (5) observed result.

Figure 2

The evolution of magnetization towards relaxation, the tip of the magnetization vector is described by the orange line.