# Detecting level crossings without solving the Hamiltonian. II. Applications to atoms and molecules

## Abstract

A number of interesting phenomena occur at points where the energy levels of an atom or a molecule (anti) cross as a function of some parameter such as an external field. In a previous paper [M. Bhattacharya and C. Raman, Phys. Rev. Lett. 97, 140405 (2006)] we have outlined powerful mathematical techniques useful in identifying the parameter values at which such (avoided) crossings occur. In the accompanying article [M. Bhattacharya and C. Raman, Phys. Rev A 75, 033405 (2007)] we have developed the mathematical basis of these algebraic techniques in some detail. In this article we apply these level-crossing methods to the spectra of atoms and molecules in a magnetic field. In the case of atoms the final result is the derivation of a class of invariants of the Breit-Rabi Hamiltonian of magnetic resonance. These invariants completely describe the parametric symmetries of the Hamiltonian. In the case of molecules we present an indicator which can tell when the Born-Oppenheimer approximation breaks down without using any information about the molecular potentials other than the fact that they are real. We frame our discussion in the context of Feshbach resonances in the atom-pair {sup 23}Na-{sup 85}Rb which are of current interest.

- Authors:

- School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332 (United States)

- Publication Date:

- OSTI Identifier:
- 20982354

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. A; Journal Volume: 75; Journal Issue: 3; Other Information: DOI: 10.1103/PhysRevA.75.033406; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS; ATOMS; BORN-OPPENHEIMER APPROXIMATION; ENERGY LEVELS; HAMILTONIANS; MAGNETIC FIELDS; MAGNETIC RESONANCE; MATHEMATICAL SOLUTIONS; MOLECULES; POTENTIAL ENERGY; POTENTIALS; RUBIDIUM 85; SODIUM 23; SPECTRA; SYMMETRY

### Citation Formats

```
Bhattacharya, M., and Raman, C.
```*Detecting level crossings without solving the Hamiltonian. II. Applications to atoms and molecules*. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVA.75.033406.

```
Bhattacharya, M., & Raman, C.
```*Detecting level crossings without solving the Hamiltonian. II. Applications to atoms and molecules*. United States. doi:10.1103/PHYSREVA.75.033406.

```
Bhattacharya, M., and Raman, C. Thu .
"Detecting level crossings without solving the Hamiltonian. II. Applications to atoms and molecules". United States.
doi:10.1103/PHYSREVA.75.033406.
```

```
@article{osti_20982354,
```

title = {Detecting level crossings without solving the Hamiltonian. II. Applications to atoms and molecules},

author = {Bhattacharya, M. and Raman, C.},

abstractNote = {A number of interesting phenomena occur at points where the energy levels of an atom or a molecule (anti) cross as a function of some parameter such as an external field. In a previous paper [M. Bhattacharya and C. Raman, Phys. Rev. Lett. 97, 140405 (2006)] we have outlined powerful mathematical techniques useful in identifying the parameter values at which such (avoided) crossings occur. In the accompanying article [M. Bhattacharya and C. Raman, Phys. Rev A 75, 033405 (2007)] we have developed the mathematical basis of these algebraic techniques in some detail. In this article we apply these level-crossing methods to the spectra of atoms and molecules in a magnetic field. In the case of atoms the final result is the derivation of a class of invariants of the Breit-Rabi Hamiltonian of magnetic resonance. These invariants completely describe the parametric symmetries of the Hamiltonian. In the case of molecules we present an indicator which can tell when the Born-Oppenheimer approximation breaks down without using any information about the molecular potentials other than the fact that they are real. We frame our discussion in the context of Feshbach resonances in the atom-pair {sup 23}Na-{sup 85}Rb which are of current interest.},

doi = {10.1103/PHYSREVA.75.033406},

journal = {Physical Review. A},

number = 3,

volume = 75,

place = {United States},

year = {Thu Mar 15 00:00:00 EDT 2007},

month = {Thu Mar 15 00:00:00 EDT 2007}

}