Symplectic structures and dynamical symmetry groups
Apart from the total energy, the two-dimensional isotropic harmonic oscillator possesses three independent constants of motion which, withthe standard symplectic structure, generates a dynamical symmetry group isomorphic to SU(2). We show that, by suitably redefining thesymplectic structure, any of...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2004 |
| País: | México |
| Institución: | Benemérita Universidad Autónoma de Puebla |
| Repositorio: | Redalyc-BUAP |
| OAI Identifier: | oai:redalyc.org:57050609 |
| Acceso en línea: | https://www.redalyc.org/articulo.oa?id=57050609 |
| Access Level: | acceso abierto |
| Palabra clave: | Física, Astronomía y Matemáticas quantization symplectic structures Dynamical symmetry groups |
| Sumario: | Apart from the total energy, the two-dimensional isotropic harmonic oscillator possesses three independent constants of motion which, withthe standard symplectic structure, generates a dynamical symmetry group isomorphic to SU(2). We show that, by suitably redefining thesymplectic structure, any of these three constants of motion can be used as a Hamiltonian, and that the remaining two, together with thetotal energy, generate a dynamical symmetry group isomorphic to SU(1,1). We also show that the standard energy levels of the quantumtwo-dimensional isotropic harmonic oscillator and their degeneracies are obtained making use of the appropriate representations of SU(1,1),provided that the canonical commutation relations are modified according to the new symplectic structure. Whereas in classical mechanicsthe different symplectic structures lead to equivalent formulations of the equations of motion, in quantum mechanics the modifications of thecommutation relations should be accompanied by modifications in the interpretation of the formalism in order to obtain results equivalent tothose found with the common relations. |
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