How to detect level crossings without looking at the spectrum

TL;DR

The paper presents a method to detect eigenvalue crossings in quantum Hamiltonians without directly computing eigenvalues, enabling analytical handling and visualization of crossings, demonstrated on the Breit-Rabi Hamiltonian.

Contribution

It introduces a novel approach to identify eigenvalue crossings analytically without spectrum calculation, enhancing understanding and visualization of avoided crossings.

Findings

Method successfully detects crossings in realistic Hamiltonians

Applicable to analytical and visual analysis of spectra

Demonstrated on the hyperfine-Zeeman structure of hydrogen

Abstract

We remind the reader that it is possible to tell if two or more eigenvalues of a matrix are equal, without calculating the eigenvalues. We then use this property to detect (avoided) crossings in the spectra of quantum Hamiltonians representable by matrices. This approach provides a pedagogical introduction to (avoided) crossings, is capable of handling realistic Hamiltonians analytically, and offers a way to visualize crossings which is sometimes superior to that provided by the spectrum. We illustrate the method using the Breit-Rabi Hamiltonian to describe the hyperfine-Zeeman structure of the ground state hydrogen atom in a uniform magnetic field.