# Communication: Wigner functions in action-angle variables, Bohr-Sommerfeld quantization, the Heisenberg correspondence principle, and a symmetrical quasi-classical approach to the full electronic density matrix

## Abstract

It is pointed out that the classical phase space distribution in action-angle (a-a) variables obtained from a Wigner function depends on how the calculation is carried out: if one computes the standard Wigner function in Cartesian variables (p, x), and then replaces p and x by their expressions in terms of a-a variables, one obtains a different result than if the Wigner function is computed directly in terms of the a-a variables. Furthermore, the latter procedure gives a result more consistent with classical and semiclassical theory - e.g., by incorporating the Bohr-Sommerfeld quantization condition (quantum states defined by integer values of the action variable) as well as the Heisenberg correspondence principle for matrix elements of an operator between such states - and has also been shown to be more accurate when applied to electronically non-adiabatic applications as implemented within the recently developed symmetrical quasi-classical (SQC) Meyer-Miller (MM) approach. Moreover, use of the Wigner function (obtained directly) in a-a variables shows how our standard SQC/MM approach can be used to obtain off-diagonal elements of the electronic density matrix by processing in a different way the same set of trajectories already used (in the SQC/MM methodology) to obtain the diagonal elements.

- Authors:

- Univ. of California, Berkeley, CA (United States). Dept. of Chemistry, Kenneth S. Pitzer Center for Theoretical Chemistry
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Chemcial Sciences Division

- Publication Date:

- Research Org.:
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)

- OSTI Identifier:
- 1379579

- Alternate Identifier(s):
- OSTI ID: 1328640

- Grant/Contract Number:
- AC02-05CH11231; CHE-1148645

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of Chemical Physics

- Additional Journal Information:
- Journal Volume: 145; Journal Issue: 8; Journal ID: ISSN 0021-9606

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

### Citation Formats

```
Miller, William H., and Cotton, Stephen J..
```*Communication: Wigner functions in action-angle variables, Bohr-Sommerfeld quantization, the Heisenberg correspondence principle, and a symmetrical quasi-classical approach to the full electronic density matrix*. United States: N. p., 2016.
Web. doi:10.1063/1.4961551.

```
Miller, William H., & Cotton, Stephen J..
```*Communication: Wigner functions in action-angle variables, Bohr-Sommerfeld quantization, the Heisenberg correspondence principle, and a symmetrical quasi-classical approach to the full electronic density matrix*. United States. doi:10.1063/1.4961551.

```
Miller, William H., and Cotton, Stephen J.. Sun .
"Communication: Wigner functions in action-angle variables, Bohr-Sommerfeld quantization, the Heisenberg correspondence principle, and a symmetrical quasi-classical approach to the full electronic density matrix". United States.
doi:10.1063/1.4961551. https://www.osti.gov/servlets/purl/1379579.
```

```
@article{osti_1379579,
```

title = {Communication: Wigner functions in action-angle variables, Bohr-Sommerfeld quantization, the Heisenberg correspondence principle, and a symmetrical quasi-classical approach to the full electronic density matrix},

author = {Miller, William H. and Cotton, Stephen J.},

abstractNote = {It is pointed out that the classical phase space distribution in action-angle (a-a) variables obtained from a Wigner function depends on how the calculation is carried out: if one computes the standard Wigner function in Cartesian variables (p, x), and then replaces p and x by their expressions in terms of a-a variables, one obtains a different result than if the Wigner function is computed directly in terms of the a-a variables. Furthermore, the latter procedure gives a result more consistent with classical and semiclassical theory - e.g., by incorporating the Bohr-Sommerfeld quantization condition (quantum states defined by integer values of the action variable) as well as the Heisenberg correspondence principle for matrix elements of an operator between such states - and has also been shown to be more accurate when applied to electronically non-adiabatic applications as implemented within the recently developed symmetrical quasi-classical (SQC) Meyer-Miller (MM) approach. Moreover, use of the Wigner function (obtained directly) in a-a variables shows how our standard SQC/MM approach can be used to obtain off-diagonal elements of the electronic density matrix by processing in a different way the same set of trajectories already used (in the SQC/MM methodology) to obtain the diagonal elements.},

doi = {10.1063/1.4961551},

journal = {Journal of Chemical Physics},

number = 8,

volume = 145,

place = {United States},

year = {Sun Aug 28 00:00:00 EDT 2016},

month = {Sun Aug 28 00:00:00 EDT 2016}

}

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