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Communications in Theoretical Physics

Properties of Coherent State for Hydrogen Atom

Xu Bo-wei, Ye Fei, Guo-Hui Ding

Published Feb 15, 2003• 2 citations
Publisher

Abstract

We construct the coherent state for the two-dimensional hydrogen atom, for which the averaged motion of position describes a Kepler ellipse. The coherent state can be expanded with respect to the eigenstates of 2D hydrogen atom, from which we evaluate the binding energy of the wave packet for this state.
DOI
10.1088/0253-6102/39/2/151

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Majumdar and Sharatchandra recently proposed a set of ``coherent states'' for the hydrogen atom [Phys. Rev. A 56, R3322 (1997)]. These states satisfy some of the typical properties of coherent states, such as, for example, continuity in the parameters. The time evolution of these states is given by the classical evolution of the angle variables, and the expectation values of the quantum observables behave quasiclassically. However, those authors also claimed that, although one does not obtain exactly the same state after a Kepler period, the gross features of their wave packets do not change. We show that this is not correct, and that these wave packets do not remain localized on the classical variables, but spread rapidly over the Keplerian orbit. The localization properties of these states do not improve in the limit of large quantum numbers.

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Theoretical and mathematical physics•Jan 1, 2021•1 citation
Majumdar and Sharatchandra recently proposed a set of ``coherent states'' for the hydrogen atom [Phys. Rev. A 56, R3322 (1997)]. These states satisfy some of the typical properties of coherent states, such as, for example, continuity in the parameters. The time evolution of these states is given by the classical evolution of the angle variables, and the expectation values of the quantum observables behave quasiclassically. However, those authors also claimed that, although one does not obtain exactly the same state after a Kepler period, the gross features of their wave packets do not change. We show that this is not correct, and that these wave packets do not remain localized on the classical variables, but spread rapidly over the Keplerian orbit. The localization properties of these states do not improve in the limit of large quantum numbers.
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 L2(S3)L^2({\mathbb S}^3)L2(S3) for the S3\mathbb S^3S3 sphere. We study the coherent states for the S3\mathbb S^3S3 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]).