Communication: Wigner functions in actionangle variables, BohrSommerfeld quantization, the Heisenberg correspondence principle, and a symmetrical quasiclassical approach to the full electronic density matrix
Abstract
It is pointed out that the classical phase space distribution in actionangle (aa) 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 aa variables, one obtains a different result than if the Wigner function is computed directly in terms of the aa variables. Furthermore, the latter procedure gives a result more consistent with classical and semiclassical theory  e.g., by incorporating the BohrSommerfeld 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 nonadiabatic applications as implemented within the recently developed symmetrical quasiclassical (SQC) MeyerMiller (MM) approach. Moreover, use of the Wigner function (obtained directly) in aa variables shows how our standard SQC/MM approach can be used to obtain offdiagonal 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) (SC22)
 OSTI Identifier:
 1379579
 Grant/Contract Number:
 AC0205CH11231; CHE1148645
 Resource Type:
 Journal Article: Accepted Manuscript
 Journal Name:
 Journal of Chemical Physics
 Additional Journal Information:
 Journal Volume: 145; Journal Issue: 8; Journal ID: ISSN 00219606
 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 actionangle variables, BohrSommerfeld quantization, the Heisenberg correspondence principle, and a symmetrical quasiclassical 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 actionangle variables, BohrSommerfeld quantization, the Heisenberg correspondence principle, and a symmetrical quasiclassical approach to the full electronic density matrix. United States. doi:10.1063/1.4961551.
Miller, William H., and Cotton, Stephen J. 2016.
"Communication: Wigner functions in actionangle variables, BohrSommerfeld quantization, the Heisenberg correspondence principle, and a symmetrical quasiclassical 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 actionangle variables, BohrSommerfeld quantization, the Heisenberg correspondence principle, and a symmetrical quasiclassical 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 actionangle (aa) 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 aa variables, one obtains a different result than if the Wigner function is computed directly in terms of the aa variables. Furthermore, the latter procedure gives a result more consistent with classical and semiclassical theory  e.g., by incorporating the BohrSommerfeld 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 nonadiabatic applications as implemented within the recently developed symmetrical quasiclassical (SQC) MeyerMiller (MM) approach. Moreover, use of the Wigner function (obtained directly) in aa variables shows how our standard SQC/MM approach can be used to obtain offdiagonal 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 = 2016,
month = 8
}
Web of Science

AharonovBohm effect in quantumtoclassical correspondence of the Heisenberg principle
The exact energy spectrum and wave function of a charged particle moving in the Coulomb field and AharonovBohm's magnetic flux are solved by the nonintegrable phase factor. The universal formula for the matrix elements of the radial operator r{sup {alpha}} of arbitrary power {alpha} is given by an analytical solution. The difference between the classical limit of matrix elements of inverse radius in quantum mechanics and the Fourier components of the corresponding quantity for the pure Coulomb system in classical mechanics is examined in reference to the correspondence principle of Heisenberg. Explicit calculation shows that the influence of nonlocal AharonovBohmmore » 
Area in phase space as determiner of transition probability: BohrSommerfeld bands, Wigner ripples, and Fresnel zones
We consider an oscillator subjected to a sudden change in equilibrium position or in effective spring constant, or bothto a squeeze in the language of quantum optics. We analyze the probability of transition from a given initial state to a final state, in its dependence on finalstate quantum number. We make use of five sources of insight: BohrSommerfeld quantization via bands in phase space, area of overlap between beforesqueeze band and aftersqueeze band, interference in phase space, Wigner function as quantum update of BS band and nearzone Fresnel diffraction as mockup Wigner function.