自链接磁场中带电粒子运动的本征波函数
Eigen Wavefunctions of a Charged Particle Moving in a Self-Linking Magnetic Field
摘要：
在本文中，我们解决了一个磁场中的单粒子薛定谔方程的磁通线呈现相互链接。为了使这个问题易于分析，我们考虑一个高对称性的情况下，粒子在一个三球$(S^3)$。矢势由两个哈密顿量$H(ř)=\hat{h}(ř) \cdot\vecσ$的Berry连接得到，其中$ř\in S^3$、$\hat{h}\in S^2$和$\vecσ$是泡利矩阵。为了产生连接的通量线，映射$\hat{h}:S^3\to S^2$具有非平凡同伦。这个问题对于一个特定的映射（$\hat{h}$）是完全可解的。得到的本征函数是SO（4）球谐函数，与无磁场时的本征函数相同。高度非平凡的磁场以一种让人想起塞曼效应的方式提升了能谱中的简并度。
英文摘要：
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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