# Stability of good quantum numbers in ground states

## TL;DR

This paper investigates the stability of good quantum numbers, which are eigenvalues of a bounded self-adjoint operator commuting with the Hamiltonian, under perturbations, using order theory in quantum mechanics.

## Contribution

It introduces a new perspective on the stability of quantum numbers by applying order theory to analyze perturbations of Hamiltonians with discrete spectra.

## Key findings

- Good quantum numbers remain stable under certain perturbations.
- Order theory provides a framework for analyzing eigenvalue stability.
- Results have implications for understanding quantum state properties under perturbations.

## Abstract

Let $H$ be a self-adjoint operator, bounded from below and let $O$ be a bounded self-adjoint operator with purely discrete spectrum. Suppose that (i) $E(H)=\inf \mathrm{spec}(H)$ is a simple eigenvalue, and (ii) $H$ strongly commutes with $O$. Let $\psi_H$ be the eigenvector associated with $E(H)$. By the assumptions (i) and (ii), $\psi_H$ is an eigenvector of $O$: $O\psi_H=\mu(H)\psi_H$. In the context of quantum mechanics, $\mu(H)$ is called a good quantum number. In this note, we examine the stability of $\mu(H)$ under perturbations of $H$ from a viewpoint of the order theory.