How to detect level crossings without looking at the spectrum

Details How to detect level crossings without looking at the spectrum How to detect level crossings without looking at the spectrum

2007-05-14

arxiv.org/abs/0705.1958v1

Physics

Atomic Physics

Accepted for publication in the American Journal of Physics

Scientific paper

10.1119/1.2757622

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.

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