dorsal/arxiv
View SchemaHeisenberg Evolution WKB and Symplectic Area Phases
| Authors | T. A. Osborn, M. F. Kondratieva |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0201029 |
| URL | https://arxiv.org/abs/quant-ph/0201029 |
| DOI | 10.1088/0305-4470/35/25/309 |
| Journal | J. Phys. A, 35:5279-5303, 2002 |
Abstract
The Schrodinger and Heisenberg evolution operators are represented in quantum phase space by their Weyl symbols. Their semiclassical approximations are constructed in the short and long time regimes. For both evolution problems, the WKB representation is purely geometrical: the amplitudes are functions of a Poisson bracket and the phase is the symplectic area of a region in phase space bounded by trajectories and chords. A unified approach to the Schrodinger and Heisenberg semiclassical evolutions is developed by introducing an extended phase space. In this setting Maslov's pseudodifferential operator version of WKB analysis applies and represents these two problems via a common higher dimensional Schrodinger evolution, but with different extended Hamiltonians. The evolution of a Lagrangian manifold in the extended phase space, defined by initial data, controls the phase, amplitude and caustic behavior. The symplectic area phases arise as a solution of a boundary condition problem. Various applications and examples are considered.
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"abstract": "The Schrodinger and Heisenberg evolution operators are represented in quantum\nphase space by their Weyl symbols. Their semiclassical approximations are\nconstructed in the short and long time regimes. For both evolution problems,\nthe WKB representation is purely geometrical: the amplitudes are functions of a\nPoisson bracket and the phase is the symplectic area of a region in phase space\nbounded by trajectories and chords. A unified approach to the Schrodinger and\nHeisenberg semiclassical evolutions is developed by introducing an extended\nphase space. In this setting Maslov\u0027s pseudodifferential operator version of\nWKB analysis applies and represents these two problems via a common higher\ndimensional Schrodinger evolution, but with different extended Hamiltonians.\nThe evolution of a Lagrangian manifold in the extended phase space, defined by\ninitial data, controls the phase, amplitude and caustic behavior. The\nsymplectic area phases arise as a solution of a boundary condition problem.\nVarious applications and examples are considered.",
"arxiv_id": "quant-ph/0201029",
"authors": [
"T. A. Osborn",
"M. F. Kondratieva"
],
"categories": [
"quant-ph"
],
"doi": "10.1088/0305-4470/35/25/309",
"journal_ref": "J. Phys. A, 35:5279-5303, 2002",
"title": "Heisenberg Evolution WKB and Symplectic Area Phases",
"url": "https://arxiv.org/abs/quant-ph/0201029"
},
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