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

TL;DR

This paper compares the trajectories of electrons in low-speed regimes using Ehrenfest's theorem within Copenhagen interpretation and Bohm's hidden-variable approach, highlighting their similarities and differences.

Contribution

It provides a comparative analysis of electron trajectories derived from Ehrenfest's theorem and Bohmian mechanics using Pauli's wave function.

Findings

Both approaches yield similar trajectories in certain conditions

Differences arise in the interpretation of electron paths and their physical meaning

The study clarifies the relationship between expectation values and Bohmian trajectories

Abstract

Electrons moving at slow speeds much lower that 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 which is the more conservative Copenhagen's interpretation denying a trajectory of the electron but allowing a trajectory to the electron expectation value through 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.