Pseudospin symmetry in supersymmetric quantum mechanics: Schr\"odinger equations

TL;DR

This paper investigates the origin and breaking mechanism of pseudospin symmetry in quantum mechanics using supersymmetry, perturbation theory, and the similarity renormalization group, providing a deeper understanding of PSS in Schrödinger equations.

Contribution

It demonstrates how pseudospin symmetry naturally emerges in SUSY partner Hamiltonians and explores its perturbative nature and energy dependence.

Findings

PSS-conserving terms appear in SUSY partner Hamiltonians.

Eigenstates of Hamiltonians are identical except for intruder states.

Pseudospin-orbit splittings decrease with increasing energy.

Abstract

The origin of pseudospin symmetry (PSS) and its breaking mechanism are explored by combining supersymmetry (SUSY) quantum mechanics, perturbation theory, and the similarity renormalization group (SRG) method. The Schr\"odinger equation is taken as an example, corresponding to the lowest-order approximation in transforming a Dirac equation into a diagonal form by using the SRG. It is shown that while the spin-symmetry-conserving term appears in the single-particle Hamiltonian $H$, the PSS-conserving term appears naturally in its SUSY partner Hamiltonian $\tilde{H}$. The eigenstates of Hamiltonians $H$ and $\tilde{H}$ are exactly one-to-one identical except for the so-called intruder states. In such a way, the origin of PSS deeply hidden in $H$ can be traced in its SUSY partner Hamiltonian $\tilde{H}$. The perturbative nature of PSS in the present potential without spin-orbit term is demonstrated by the perturbation calculations, and the PSS-breaking term can be regarded as a very small perturbation on the exact PSS limits. A general tendency that the pseudospin-orbit splittings become smaller with increasing single-particle energies can also be interpreted in an explicit way.