|
| related topics |
| {equation, function, exp} |
| {energy, gaussian, time} |
| {group, space, representation} |
| {cos, sin, state} |
| {classical, space, random} |
| {wave, scattering, interference} |
| {time, decoherence, evolution} |
| {time, wave, function} |
|
Wigner functions, contact interactions, and matching
Mark A. Walton
abstract: Quantum mechanics in phase space (or deformation quantization) appears to
fail as an autonomous quantum method when infinite potential walls are present.
The stationary physical Wigner functions do not satisfy the normal eigen
equations, the *-eigen equations, unless an ad hoc boundary potential is added
[Dias-Prata]. Alternatively, they satisfy a different, higher-order,
``*-eigen-* equation'', locally, i.e. away from the walls [Kryukov-Walton].
Here we show that this substitute equation can be written in a very simple
form, even in the presence of an additional, arbitrary, but regular potential.
The more general applicability of the -eigen- equation is then demonstrated.
First, using an idea from [Fairlie-Manogue], we extend it to a dynamical
equation describing time evolution. We then show that also for general contact
interactions, the -eigen- equation is satisfied locally. Specifically, we treat
the most general possible (Robin) boundary conditions at an infinite wall,
general one-dimensional point interactions, and a finite potential jump.
Finally, we examine a smooth potential, that has simple but different
expressions for x positive and negative. We find that the -eigen- equation is
again satisfied locally. It seems, therefore, that the -eigen- equation is
generally relevant to the matching of Wigner functions; it can be solved
piece-wise and its solutions then matched.
- oai_identifier:
- oai:arXiv.org:quant-ph/0609213
- categories:
- quant-ph
- comments:
- 20 pages, no figures
- doi:
- 10.1016/j.aop.2006.11.015
- arxiv_id:
- quant-ph/0609213
- journal_ref:
- Ann. Phys.322:2233-2248, 2007
- created:
- 2006-09-27
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