Symplectic structures and dynamical symmetry groups
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| Format: | Artículo científico |
| Language: | en |
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Sociedad Mexicana de Física A.C.
2004
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| _version_ | 1876426686985142272 |
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| author | G. F. Torres del Castillo |
| author_facet | G. F. Torres del Castillo |
| contents | Symplectic structures and dynamical symmetry groups G. F. Torres del Castillo M.P. Velázquez Quesada Física, Astronomía y Matemáticas quantization symplectic structures 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 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. 2004 artículo científico 0035-001X https://www.redalyc.org/articulo.oa?id=57050609 en http://www.redalyc.org/revista.oa?id=570 Revista Mexicana de Física application/pdf Sociedad Mexicana de Física A.C. Revista Mexicana de Física (México) Num.Re6 Vol.50 |
| format | Artículo científico |
| id | redalyc_57050609 |
| institution | Redalyc |
| language | en |
| publishDate | 2004 |
| publisher | Sociedad Mexicana de Física A.C. |
| spellingShingle | Symplectic structures and dynamical symmetry groups G. F. Torres del Castillo Física, Astronomía y Matemáticas quantization symplectic structures Dynamical symmetry groups Symplectic structures and dynamical symmetry groups G. F. Torres del Castillo M.P. Velázquez Quesada Física, Astronomía y Matemáticas quantization symplectic structures 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 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. 2004 artículo científico 0035-001X https://www.redalyc.org/articulo.oa?id=57050609 en http://www.redalyc.org/revista.oa?id=570 Revista Mexicana de Física application/pdf Sociedad Mexicana de Física A.C. Revista Mexicana de Física (México) Num.Re6 Vol.50 |
| title | Symplectic structures and dynamical symmetry groups |
| topic | Física, Astronomía y Matemáticas quantization symplectic structures Dynamical symmetry groups |
| url | https://www.redalyc.org/articulo.oa?id=57050609 |