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Coherent states for the hydrogen atom: discrete and continuous spectra

Semyon Pol'shin

Published Aug 9, 2001• 4 citations
OpenAlex

Abstract

We construct the systems of generalised coherent states for the discrete and continuous spectra of the hydrogen atom. These systems are expressed in terms of 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 interwines irreducible representations of the $SO(4, 2)$ group.

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We construct the systems of generalised coherent states for the discrete and continuous spectra of the hydrogen atom. These systems are expressed in terms of elementary functions and are invariant under the SO(3,2)SO(3, 2)SO(3,2) (discrete spectrum) and SO(4,1)SO(4, 1)SO(4,1) (continuous spectrum) subgroups of the dynamical symmetry group SO(4,2)SO(4, 2)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)SO(4, 2)SO(4,2) group.
A new approach is developed for solving spectral problems for operators with continuous spectrum, which consists in the integral transform of the problem by using coherent Schwartz distributions. The constructed family of coherent distributions is a complete analogue of the family of ordinary coherent states. More precisely, it satisfies all Gazeau–Klauder axioms satisfied by the usual coherent states. But in contrast to the coherent states belonging to the point spectrum of the annihilation operator (or operators), the coherent distributions belong to the continuous spectrum of some Hermitian operators. Therefore, the coherent distributions work better than the coherent states as the kernel of the integral representation of generalized eigenfunctions of operators with continuous spectrum. In this work, this approach is demonstrated with an example of solving a basic problem of quantum mechanics, i.e., the problem of the continuous part of the spectrum of the Hamiltonian of the hydrogen atom.
Unitary operator coherent states, as defined by Klauder (1963), Perelomov (1972) and Gilmore (1974), are considered for the harmonic series irreducible representations of U(p,q). Various properties of these states are reviewed, including their reproducing kernel and measure explicit forms, as well as the representation they lead to for the U(p,q) generators. Starting from a contraction of the su(p,q) Lie algebra, both Dyson and Holstein-Primakoff boson realisations of u(p,q) are obtained in terms of pq pairs of boson creation, annihilation operators and of the generators of a U(p)*U(q) intrinsic group. Such boson realisations are applied to determine the matrix elements of the U(p,q) generators between discrete bases classified according to the chain U(p,q) contains/implies U(p)*U(q). Finally, the isomorphism between so(4,2) and su(2,2) is employed to derive the corresponding properties of the SO(4,2) unitary operator coherent states for application in atomic physics.