Spin-dependent Bohm trajectories for Pauli and Dirac eigenstates of hydrogen

TL;DR

This paper applies the de Broglie-Bohm causal interpretation to hydrogen atom eigenstates, incorporating spin and relativistic effects, and finds circular electron trajectories consistent across different quantum frameworks.

Contribution

It extends Bohmian trajectory analysis to spin-dependent and relativistic hydrogen eigenstates, comparing Dirac and Pauli results with previous Schrödinger-based findings.

Findings

Electron trajectories are circular around the z-axis in eigenstates.

Revolution rates for n=1 and n=2 states are computed and match across models.

Results agree with earlier Schrödinger equation analyses.

Abstract

The de Broglie-Bohm causal theory of quantum mechanics is applied to the hydrogen atom in the fully spin-dependent and relativistic framework of the Dirac equation, and in the nonrelativistic but spin-dependent framework of the Pauli equation. Eigenstates are chosen which are simultaneous eigenstates of the energy H, total angular momentum M, and z component of the total angular momentum M_z. We find the trajectories of the electron, and show that in these eigenstates, motion is circular about the z-axis, with constant angular velocity. We compute the rates of revolution for the ground (n=1) state and the n=2 states, and show that there is agreement in the relevant cases between the Dirac and Pauli results, and with earlier results on the Schrodinger equation.