Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Two simple models derived from a quantum-mechanical particle on an elliptical path

Francisco M. Fernández

TL;DR

This work analyzes two simple quantum models of a particle constrained to move on an ellipse. It employs perturbation theory and Rayleigh-Ritz computations, organized by the $D_{2}$/$C_{2v}$ symmetry, to compare a non-Hermitian Hamiltonian that is isospectral to a Hermitian partner with a Hermitian variant that breaks degeneracy when $\xi\neq0$. The authors find a robust twofold degeneracy and an exact ground state in the non-Hermitian case, with a conjectured spectrum $E_n=n^2 E_1$, while the Hermitian variant shows level splitting that grows with the order of perturbation (appearing at order $n$ for the $n$-th level). These results illustrate how symmetry and operator structure govern spectra in constrained quantum systems and demonstrate effective numerical techniques for analyzing such models.

Abstract

We analyze two simple models derived from a quantum-mechanical particle on an elliptical path. The first Hamiltonian operator is non-Hermitian but isomorphic to an Hermitian operator. It appears to exhibit the same two-fold degeneracy as the particle on a circular path. More precisely, $E_n=n^2E_1,\ n=1,2,\ldots$ (in addition to an exact eigenvalue $E_0=0$). The second Hamiltonian operator is Hermitian and does not exhibit such degeneracy. In this case the nth excited energy level splits at the nth order of perturbation theory. Both models can be described in terms of the same point-group symmetry.

Two simple models derived from a quantum-mechanical particle on an elliptical path

TL;DR

This work analyzes two simple quantum models of a particle constrained to move on an ellipse. It employs perturbation theory and Rayleigh-Ritz computations, organized by the / symmetry, to compare a non-Hermitian Hamiltonian that is isospectral to a Hermitian partner with a Hermitian variant that breaks degeneracy when . The authors find a robust twofold degeneracy and an exact ground state in the non-Hermitian case, with a conjectured spectrum , while the Hermitian variant shows level splitting that grows with the order of perturbation (appearing at order for the -th level). These results illustrate how symmetry and operator structure govern spectra in constrained quantum systems and demonstrate effective numerical techniques for analyzing such models.

Abstract

We analyze two simple models derived from a quantum-mechanical particle on an elliptical path. The first Hamiltonian operator is non-Hermitian but isomorphic to an Hermitian operator. It appears to exhibit the same two-fold degeneracy as the particle on a circular path. More precisely, (in addition to an exact eigenvalue ). The second Hamiltonian operator is Hermitian and does not exhibit such degeneracy. In this case the nth excited energy level splits at the nth order of perturbation theory. Both models can be described in terms of the same point-group symmetry.

Paper Structure

This paper contains 6 sections, 26 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. The model
  3. Perturbation theory
  4. Rayleigh-Ritz method
  5. Alternative model
  6. Conclusions

Figures (3)

  • Figure 1: RPM (blue circles), PT (solid red line) and improved PT (dashed red line) eigenvalues with $n=1,2,3,4$ of model (\ref{['eq:Schro']})
  • Figure 2: Lowest RPM eigenvalues (blue circles and red squares) and improved PT eigenvalues (solid green line) of model (\ref{['eq:H_2']})
  • Figure 3: RPM eigenvalues (blue circles and red squares) and PT series (blue solid and red dashed lines) for the first and second excited states of model (\ref{['eq:H_2']})