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

Dah-Wei Chiou·Dung-Hai Lee·Wu-Yi Hsiang
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Abstract

In this paper we solve the one-particle Schrödinger 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 $H(ř)=\hat{h}(ř) \cdot\vecσ$, where $ř\in S^3$, $\hat{h}\in S^2$ and $\vecσ$ 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.

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Dah-Wei Chiou, Dung-Hai Lee, Wu-Yi Hsiang. 2004-11-11. Eigen Wavefunctions of a Charged Particle Moving in a Self-Linking Magnetic Field. https://arxiv.org/abs/math-ph/0411044

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