How to detect level crossings without looking at the spectrum
Atomic Physics
2009-11-13 v1
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.
Keywords
Cite
@article{arxiv.0705.1958,
title = {How to detect level crossings without looking at the spectrum},
author = {M. Bhattacharya},
journal= {arXiv preprint arXiv:0705.1958},
year = {2009}
}