arXiv:quant-ph/0108042 · Journal of Physics A Mathematical and General

Coherent states for the hydrogen atom: discrete and continuous spectra

S A Pol'shin

Published Dec 5, 2001• 3 citations

Abstract

We construct the systems of generalized coherent states for the discrete and continuous spectra of the hydrogen atom. These systems are expressed in elementary functions and are invariant under the SO (3, 2) (discrete spectrum) and SO (4, 1) (continuous spectrum) subgroups of the dynamical symmetry group SO (4, 2) of the hydrogen atom. Both systems of coherent states are particular cases of the kernel of integral operator which intertwines irreducible representations of the SO (4, 2) group.

DOI: 10.1088/0305-4470/34/49/323

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arXiv:quant-ph/0108042 · Journal of Physics A Mathematical and General

Coherent states for the hydrogen atom: discrete and continuous spectra

S A Pol'shin

Published Dec 5, 2001• 3 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/0305-4470/34/49/323

Related papers

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The aim of this chapter is to present a construction of a set of coherent states for the hydrogen atom proposed by Villegas-Blas (The Laplacian on the n-sphere, the hydrogen atom and the Bargmann space representation. Ph.D. thesis, Mathematics Department, University of Virginia, 1996, [251]), Thomas and Villegas-Blas (Commun Math Phys 187:623–645, 1997, [244]). We show that in a semiclassical sense they concentrate essentially around the Kepler orbits (in configuration space) of the classical motion. A suitable unitary transformation (the Fock operator) maps the pure point subspace of the hydrogen atom Hamitonian onto the Hilbert space $L^2({\mathbb S}^3)$ for the $\mathbb S^3$ sphere. We study the coherent states for the $\mathbb S^3$ sphere (as introduced by Uribe (J Funct Anal 59:535–556, 1984, [247]) and show that the action of the group SO(4) is irreducible in the space generated by the spherical harmonics of a given degree. Note that coherent states for the hydrogen atom have been extensively studied by J. Klauder and his school. We have chosen not to present them here and refer the interested reader to Klauder and Skagerstam (Coherent states, Worlds Scientific Publishing, Singapore, 1985, [167]).The aim of this chapter is to present a construction of a set of coherent states for the hydrogen atom proposed by Villegas-Blas (The Laplacian on the n-sphere, the hydrogen atom and the Bargmann space representation. Ph.D. thesis, Mathematics Department, University of Virginia, 1996, [251]), Thomas and Villegas-Blas (Commun Math Phys 187:623–645, 1997, [244]). We show that in a semiclassical sense they concentrate essentially around the Kepler orbits (in configuration space) of the classical motion. A suitable unitary transformation (the Fock operator) maps the pure point subspace of the hydrogen atom Hamitonian onto the Hilbert space $L^2({\mathbb S}^3)$ for the $\mathbb S^3$ sphere. We study the coherent states for the $\mathbb S^3$ sphere (as introduced by Uribe (J Funct Anal 59:535–556, 1984, [247]) and show that the action of the group SO(4) is irreducible in the space generated by the spherical harmonics of a given degree. Note that coherent states for the hydrogen atom have been extensively studied by J. Klauder and his school. We have chosen not to present them here and refer the interested reader to Klauder and Skagerstam (Coherent states, Worlds Scientific Publishing, Singapore, 1985, [167]).

Theoretical and mathematical physicsJan 1, 20211 citation

SO(3, 2)

(discrete spectrum) and

SO(4, 1)

(continuous spectrum) subgroups of the dynamical symmetry group

SO(4, 2)

of the hydrogen atom. Both systems of coherent states are particular cases of the kernel of integral operator which interwines irreducible representations of the

SO(4, 2)

group.

L^2({\mathbb S}^3)

for the

\mathbb S^3

sphere. We study the coherent states for the

\mathbb S^3

sphere (as introduced by Uribe (J Funct Anal 59:535–556, 1984, [247]) and show that the action of the group SO(4) is irreducible in the space generated by the spherical harmonics of a given degree. Note that coherent states for the hydrogen atom have been extensively studied by J. Klauder and his school. We have chosen not to present them here and refer the interested reader to Klauder and Skagerstam (Coherent states, Worlds Scientific Publishing, Singapore, 1985, [167]).