On the crossing of the energy levels of a parameter-dependent quantum-mechanical Hamiltonian

TL;DR

This paper revisits the non-crossing rule for energy levels in parameter-dependent quantum Hamiltonians, identifies a flaw in the common proof, and explores avoided crossings and exceptional points through simple models.

Contribution

It reveals a flaw in the traditional proof of the non-crossing rule and clarifies the relationship between avoided crossings and exceptional points using simple models.

Findings

Identifies a flaw in the standard proof of the non-crossing rule.

Shows the connection between avoided crossings and exceptional points.

Uses simple models to illustrate key concepts.

Abstract

The non-crossing rule for the energy levels of a parameter-dependent Hamiltonian is revisited and a flaw in a commonly accepted proof is revealed. Some aspects of avoided crossings are illustrated by means of simple models. One of them shows the close relationship between avoided crossings and exceptional points.