Eigen Wavefunctions of a Charged Particle Moving in a Self-Linking Magnetic Field

TL;DR

This paper analytically solves the Schrödinger equation for a charged particle in a complex magnetic field with linked flux lines on a three-sphere, revealing how such fields lift degeneracies similarly to the Zeeman effect.

Contribution

It introduces an exactly solvable model of a charged particle in a linked flux magnetic field on a three-sphere, connecting topology with quantum eigenstates.

Findings

Eigenfunctions are SO(4) spherical harmonics.

Magnetic field lifts degeneracy in energy spectrum.

Eigenstates resemble those without magnetic field.

Abstract

In this paper we solve the one-particle Schr\"{o}dinger equation in a magnetic field whose flux lines exhibit mutual linking. To make this problem analytically tractable, we consider a high-symmetry situation where the particle moves in a three-sphere $(S^3)$. The vector potential is obtained from the Berry connection of the two by two Hamiltonian $H(\v{r})=\hat{h}(\v{r}) \cdot\vec{\sigma}$, where $\v{r}\in S^3$, $\hat{h}\in S^2$ and $\vec{\sigma}$ are the Pauli matrices. In order to produce linking flux lines, the map $\hat{h}:S^3\to S^2$ is made to possess nontrivial homotopy. The problem is exactly solvable for a particular mapping ($\hat{h}$) . The resulting eigenfunctions are SO(4) spherical harmonics, the same as those when the magnetic field is absent. The highly nontrivial magnetic field lifts the degeneracy in the energy spectrum in a way reminiscent of the Zeeman effect.