How to detect level crossings without looking at the spectrum

Abstract

It is possible to tell if two or more eigenvalues of a matrix are equal without calculating the eigenvalues. We use this property to detect (avoided) crossings in the spectra of 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 that is sometimes superior to that provided by the spectrum directly. 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.