On separable Pauli equations

TL;DR

This paper classifies electromagnetic potentials that allow the Pauli equation for a spin-1/2 particle to be separated into simpler equations, revealing conditions for separability and providing an algorithm to construct such systems.

Contribution

It introduces a complete classification of vector-potentials enabling separability of the Pauli equation and establishes its relation to Schrödinger equations, including an algorithm for constructing these systems.

Findings

Eleven classes of vector-potentials enable separability.

Separable Pauli equations are equivalent to two uncoupled Schrödinger equations.

Conditions for electromagnetic fields to allow separability and satisfy Maxwell's equations.

Abstract

We classify (1+3)-dimensional Pauli equations for a spin-1/2 particle interacting with the electro-magnetic field, that are solvable by the method of separation of variables. As a result, we obtain the eleven classes of vector-potentials of the electro-magnetic field A(t,x) providing separability of the corresponding Pauli equations. It is established, in particular, that the necessary condition for the Pauli equation to be separable into second-order matrix ordinary differential equations is its equivalence to the system of two uncoupled Schroedinger equations. In addition, the magnetic field has to be independent of spatial variables. We prove that coordinate systems and the vector-potentials of the electro-magnetic field providing the separability of the corresponding Pauli equations coincide with those for the Schroedinger equations. Furthermore, an efficient algorithm for constructing all coordinate systems providing the separability of Pauli equation with a fixed vector-potential of the electro-magnetic field is developed. Finally, we describe all vector-potentials A(t,x) that (a) provide the separability of Pauli equation, (b) satisfy vacuum Maxwell equations without currents, and (c) describe non-zero magnetic field.