| A. Ciências Exatas e da Terra - 3. Física - 5. Física das Partículas Elementares e Campos |
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| ON THE HYDROGEN ATOM VIA WIGNER-HEISENBERG ALGEBRA |
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Rafael de Lima Rodrigues 1, 2
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1. Universidade Federal de Campina Grande-Campus de Cuité-PB
2. Centro Brasileiro de Pesquisas Físicas-CBPF-Rio de Janeiro-RJ
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| INTRODUÇÃO: |
| The R-deformed Heisenberg or Wigner-Heisenberg (WH) algebraic technique which was recently super-realized for
quantum oscillators (Jayaraman-Rodrigues (1990) is related to the paraboson relations introduced by Green. Let us now point out that the WH algebra is given by following
(anti-)commutation relations ({A,B}=AB+BA and [A,B] = AB-BA):
H=1/2 {a-, a+}, [H, a+]=a+, [H, a-]=-a-, [a-, a+]=1 + cR,
where c is a real constant associated to the Wigner parameter and the R operator satisfies {R, a+-}=0, {R, a-}=0 and R2=1. Note that when c=0 we have the standard Heisenberg algebra.
The generalized quantum condition given in above has been
found relevant in the context of integrable models. Furthermore, this algebra was also used to solve the energy eigenvalue and eigenfunctions of the Calogero interaction, in the context of one-dimensional many-body integrable systems, in terms of a new set of phase space variables involving exchanged operators. From this WH algebra a new kind of deformed calculus has been developed. The WH algebra has been considered for the three-dimensional non-canonical oscillator to generate a representation of the orthosympletic Lie superalgebra osp(3/2), and recently Palev have investigated the 3D Wigner oscillator under a discrete non-commutative context. Also, the connection of the WH algebra with the Lie superalgebra has been studied in a detailed manner. Recently, the relevance of relations to quantization in fractional dimension (Matos (2001, 2004) has been also discussed and the properties of Weyl-ordered polynomials in operators P and Q, in fractional-dimensional quantum mechanics have been developed .
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| METODOLOGIA: |
| The Kustaanheimo-Stiefel mapping of a constrained isotropic oscillator in four dimensions (4D) onto the corresponding system in 3D yields the hydrogen atom that has been exactly solved and well-studied in the literature. (See for example,
Chen, Cornish (1984), Chen and Kibler (1985), D'Hoker and Vinet (1985.) Kostelecky, Nieto and Truax (1985} have studied in a detailed manner the relation of the supersymmetric (SUSY) Coulombian problem (1984), Amado (1988), Lange (1991), Tangerman (1993) in D-dimensions with that of SUSY isotropic oscillators in D-dimensions in the radial version. (See also Lahiri et. al. (1990). For the mapping with 3D radial oscillators, see also Bergmann and Frishman (1965}, Cahill (1990) and J. - L. Chen et. al. (2000). The connection of the D-dimensional hydrogen atom with the D-dimensional harmonic oscillator in terms of the su(1,1) algebra has been investigated by Gao-Jian Zeng al. (1994}. However, the correspondence mapping of a 4D isotropic constrained super Wigner oscillator (for super Wigner oscillators see our previews work (1994) with the corresponding super system in 3D such that the usual 3D hydrogen atom emerges in the 4D—>3D mapping in the bosonic sector has not been studied in the literature; the objectives of the present work are to do such a mapping and to analyze in detail the consequences.
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| RESULTADOS: |
| In this work, the stationary states of the hydrogen atom are mapped onto the super-Wigner oscillator by using the Kustaanheimo-Stiefel transformation.
We extend the usual Kustaanheimo-Stiefel 4D—>3D mapping to study and discuss constrained super-Wigner oscillator in four dimensions. We show that the physical hydrogen atom is the system that emerges in the bosonic sector of the mapped super 3D system. See also the preprint Notas de Física do CBPF-2008, NF0008/08, by the author of this work, www.cbpf.br.
This research was supported in part by CNPq (Brazilian Research Agency).
This work was initiated in collaboration with Jambunatha Jayaraman
(In memory), whose advises and encouragement were fundamental.
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| CONCLUSÃO: |
| In this work, we have deduced the energy eigenvalues and
eigenfunctions of the hydrogen atom via Wigner-Heisenberg (WH)
algebra in non-relativistic quantum mechanics. Indeed, from the
ladder operators for the 4-dimensional (4D) super Wigner system,
ladder operators for the mapped super 3D system, and hence for
hydrogen-like atom in bosonic sector, are deduced. The
complete spectrum for the hydrogen atom is found with considerable
simplicity. Therefore, the solutions of the time-independent
Schrödinger equation for the hydrogen atom were mapped onto the
super Wigner harmonic oscillator in 4D by using the
Kustaanheimo-Stiefel transformation.
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| Instituição de Fomento: MCT-CNPq |
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| Palavras-chave: quantum mechanics, Wigner-Heisenberg algebra, hydrogen atom. |