On the SUSY structure of spherically symmetric Pauli Hamiltonians

TL;DR

The paper addresses how to realize supersymmetric quantum mechanics in the non-relativistic Pauli Hamiltonian for a spin-$\tfrac{1}{2}$ electron under spherical symmetry. By restricting to subspaces of fixed total angular momentum $j$, it uses the spin-orbit operator $K$ as a grading (Witten parity) and constructs a radial supercharge $Q=(\boldsymbol{\sigma}\cdot\mathbf{v})$ with an appropriate vector $\mathbf{v}$ to obtain $H=Q^2$, thereby revealing a hidden SUSY structure. It recovers known radial SUSY models (free particle and Coulomb) and introduces new exactly solvable cases, including the Pauli oscillator and a Coulomb-like system with a linear superpotential, some exhibiting a mixed-type shape invariance that translates and scales parameters. This framework extends SUSY Pauli systems in three dimensions, highlights the role of the spin-orbit operator in organizing SUSY, and suggests avenues for discovering further solvable spin-orbit–coupled quantum problems.

Abstract

It is shown that the quantum Hamiltonian characterising a non-relativistic electron under the influence of an external spherical symmetric electromagnetic potential exhibits a supersymmetric structure. Both cases, spherical symmetric scalar potentials and spherical symmetric vector potentials are discussed in detail. The current approach, which includes the spin-1/2 degree of freedom, provides new insights to known models like the radial harmonic oscillator and the Coulomb problem. We also find a few new exactly solvable models, one of them exhibiting a new mixed type of shape invariance containing translation and scaling of potential parameters. The fundamental role as Witten parity played by the spin-orbit operator is high-lighted.

Figures (3)

• Figure 1: The partner potentials \ref{['V12a']} for $2m=1=\gamma$ and $\ell=2$. The left graph shows the pair $V_\pm^{(1)}$ associated with unbroken SUSY and the right graph presents the broken SUSY pair $V_\pm^{(2)}$.
• Figure 2: The partner potentials $V_\pm^{(1)}$ for $2m=1=\gamma$, $\ell=2$ and large values of the exponent $a\in[10,30 ]$. The left graph shows $V_+^{(1)}$ indicating that the limit $a\to\infty$ simulates a Neumann boundary condition at $r=1$. The left graph shows $V_-^{(1)}$ indicating a Dirichlet boundary condition at $r=1$.
• Figure 3: The probability density corresponding to the zero-energy ground state \ref{['psizeroa']} for the same parameters as in figure 2. We clearly see a localisation at $r=1$ for large $a$ as expected.