# Semiquantum versus semiclassical mechanics for simple nonlinear systems

## Abstract

Quantum mechanics has been formulated in phase space, with the Wigner function as the representative of the quantum density operator, and classical mechanics has been formulated in Hilbert space, with the Groenewold operator as the representative of the classical Liouville density function. Semiclassical approximations to the quantum evolution of the Wigner function have been defined, enabling the quantum evolution to be approached from a classical starting point. Now analogous semiquantum approximations to the classical evolution of the Groenewold operator are defined, enabling the classical evolution to be approached from a quantum starting point. Simple nonlinear systems with one degree of freedom are considered, whose Hamiltonians are polynomials in the Hamiltonian of the simple harmonic oscillator. The behavior of expectation values of simple observables and of eigenvalues of the Groenewold operator are calculated numerically and compared for the various semiclassical and semiquantum approximations.

- Authors:

- Department of Mathematics, University of Queensland, Brisbane 4072, Queensland (Australia)

- Publication Date:

- OSTI Identifier:
- 20786636

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. A; Journal Volume: 73; Journal Issue: 1; Other Information: DOI: 10.1103/PhysRevA.73.012104; (c) 2006 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; BOLTZMANN-VLASOV EQUATION; CLASSICAL MECHANICS; DEGREES OF FREEDOM; DENSITY; EIGENFUNCTIONS; EIGENVALUES; HAMILTONIANS; HARMONIC OSCILLATORS; HILBERT SPACE; NONLINEAR PROBLEMS; PHASE SPACE; POLYNOMIALS; QUANTUM MECHANICS; SEMICLASSICAL APPROXIMATION; WIGNER DISTRIBUTION

### Citation Formats

```
Bracken, A. J., and Wood, J. G..
```*Semiquantum versus semiclassical mechanics for simple nonlinear systems*. United States: N. p., 2006.
Web. doi:10.1103/PHYSREVA.73.0.

```
Bracken, A. J., & Wood, J. G..
```*Semiquantum versus semiclassical mechanics for simple nonlinear systems*. United States. doi:10.1103/PHYSREVA.73.0.

```
Bracken, A. J., and Wood, J. G.. Sun .
"Semiquantum versus semiclassical mechanics for simple nonlinear systems". United States.
doi:10.1103/PHYSREVA.73.0.
```

```
@article{osti_20786636,
```

title = {Semiquantum versus semiclassical mechanics for simple nonlinear systems},

author = {Bracken, A. J. and Wood, J. G.},

abstractNote = {Quantum mechanics has been formulated in phase space, with the Wigner function as the representative of the quantum density operator, and classical mechanics has been formulated in Hilbert space, with the Groenewold operator as the representative of the classical Liouville density function. Semiclassical approximations to the quantum evolution of the Wigner function have been defined, enabling the quantum evolution to be approached from a classical starting point. Now analogous semiquantum approximations to the classical evolution of the Groenewold operator are defined, enabling the classical evolution to be approached from a quantum starting point. Simple nonlinear systems with one degree of freedom are considered, whose Hamiltonians are polynomials in the Hamiltonian of the simple harmonic oscillator. The behavior of expectation values of simple observables and of eigenvalues of the Groenewold operator are calculated numerically and compared for the various semiclassical and semiquantum approximations.},

doi = {10.1103/PHYSREVA.73.0},

journal = {Physical Review. A},

number = 1,

volume = 73,

place = {United States},

year = {Sun Jan 15 00:00:00 EST 2006},

month = {Sun Jan 15 00:00:00 EST 2006}

}