dorsal/arxiv
View SchemaImplications of invariance of the Hamiltonian under canonical transformations in phase space
| Authors | E. D. Davis, G. I. Ghandour |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9905002 |
| URL | https://arxiv.org/abs/quant-ph/9905002 |
| DOI | 10.1088/0305-4470/35/28/307 |
| Journal | J.Phys.A35:5875-5891,2002 |
Abstract
We observe that, within the effective generating function formalism for the implementation of canonical transformations within wave mechanics, non-trivial canonical transformations which leave invariant the form of the Hamilton function of the classical analogue of a quantum system manifest themselves in an integral equation for its stationary state eigenfunctions. We restrict ourselves to that subclass of these dynamical symmetries for which the corresponding effective generating functions are necessaarily free of quantum corrections. We demonstrate that infinite families of such transformations exist for a variety of familiar conservative systems of one degree of freedom. We show how the geometry of the canonical transformations and the symmetry of the effective generating function can be exploited to pin down the precise form of the integral equations for stationary state eigenfunctions. We recover several integral equations found in the literature on standard special functions of mathematical physics. We end with a brief discussion (relevant to string theory) of the generalization to scalar field theories in 1+1 dimensions.
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"abstract": "We observe that, within the effective generating function formalism for the\nimplementation of canonical transformations within wave mechanics, non-trivial\ncanonical transformations which leave invariant the form of the Hamilton\nfunction of the classical analogue of a quantum system manifest themselves in\nan integral equation for its stationary state eigenfunctions. We restrict\nourselves to that subclass of these dynamical symmetries for which the\ncorresponding effective generating functions are necessaarily free of quantum\ncorrections. We demonstrate that infinite families of such transformations\nexist for a variety of familiar conservative systems of one degree of freedom.\nWe show how the geometry of the canonical transformations and the symmetry of\nthe effective generating function can be exploited to pin down the precise form\nof the integral equations for stationary state eigenfunctions. We recover\nseveral integral equations found in the literature on standard special\nfunctions of mathematical physics. We end with a brief discussion (relevant to\nstring theory) of the generalization to scalar field theories in 1+1\ndimensions.",
"arxiv_id": "quant-ph/9905002",
"authors": [
"E. D. Davis",
"G. I. Ghandour"
],
"categories": [
"quant-ph"
],
"doi": "10.1088/0305-4470/35/28/307",
"journal_ref": "J.Phys.A35:5875-5891,2002",
"title": "Implications of invariance of the Hamiltonian under canonical transformations in phase space",
"url": "https://arxiv.org/abs/quant-ph/9905002"
},
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