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arXiv:quant-ph/9810044

Coherent States for Discrete Spectrum Dynamics

John R. Klauder

Published Oct 14, 1998• 6 citations
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Abstract

Coherent states for general systems with discrete spectrum, such as the bound states of the hydrogen atom, are discussed. The states in question satisfy: (1) continuity of labeling, (2) resolution of unity, (3) temporal stability, and (4) an action identity. This set of reasonable physical requirements uniquely specify coherent states for the (bound state portion of the) hydrogen atom.

DOI
10.48550/arxiv.quant-ph/9810044

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Coherent states for general systems with discrete spectrum, such as the bound states of the hydrogen atom, are discussed. The states in question satisfy: (1) continuity of labeling, (2) resolution of unity, (3) temporal stability, and (4) an action identity. This set of reasonable physical requirements uniquely specify coherent states for the (bound state portion of the) hydrogen atom.
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The long-standing problem of finding coherent states for the (bound state portion of the) hydrogen atom is positively resolved. The states in question: (i) are normalized and parameterized continuously, (ii) admit a resolution of unity with a positive measure, and (iii) enjoy the property that the temporal evolution of any coherent state by the hydrogen atom Hamiltonian remains a coherent state for all time. Harmonic-oscillator coherent states In modern terms, the distinguished set of states Schrodinger introduced for the harmonic oscillator 1 are now commonly known as coherent states 2 and are given, for all z 2 I C, by jzi j e (za y \\Gammaz a) j0i = e \\Gamma 1 2 jzj 2 1 X n=0 z n p n! jni; (1) where [a; a y ] = 1 as usual. Here jni; n = 0; 1; 2; : : :, denote normalized eigenstates of the number operator N , N jni = njni, which may be identified with the eigenstates of the harmonic oscillator Hamiltonian H 0 = !N = !a y a (¯h = 1). As such it follows that ...
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.
The original canonical coherent states could be defined in several ways. As applications for other sets of coherent states arose, the rules of definition were correspondingly changed. Among such rule changes were a change of group and relaxation of the analytic nature of the labels. Recent developments have done away with the group connections altogether and thereby allowed sets of coherent states to be defined that are temporally stable for a wide variety of dynamical systems including the hydrogen atom. This article outlines some of the current trends in the definitions and properties of present-day coherent states.