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arXiv:quant-ph/0110108

The Current State of Coherent States

John R. Klauder

Published Oct 17, 2001• 29 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0110108

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For a system with one degree of freedom, coherent states that are parametrized by classical canonical action-angle variables are introduced. These states also possess continuity of labelling, a resolution of unity, and temporal stability. The insistence on canonical action-angle variables strongly restricts any remaining arbitrariness in the coherent state definition. Such states are introduced for semibounded Hamiltonian operators having either a discrete or a continuous spectrum. Hamiltonians that have both discrete and continuous parts in their spectrum are also discussed.

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Comment on “Coherent states for the hydrogen atom”

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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.

Physical Review A•Jan 1, 1999•10 citations

Coherent States for the Hydrogen Atom

John R. Klauder

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 ...

•Nov 23, 1995•105 citations
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.
Klauder's recent generalization of the harmonic oscillator coherent states [J. Phys A 29, L293 (1996)], is applicable only in nondegenerate systems, requiring some additional structure if applied to systems with degeneracies. The author suggests how this structure could be added, and applies the complete method to the hydrogen atom problem. To illustrate how a certain degree of freedom in the construction may be exercised, states are constructed, which are initially localized and evolve semiclassically, and whose long time evolution exhibits ``fractional revivals.''
The study of coherent states (CS) for a quantum mechanical system has \nreceived a lot of attention. The definition, applications, generalizations of \nsuch states have been the subject of work by researchers. A common starting \npoint of all these approaches is the observation of properties of the original \nCS for the harmonic oscillator. It is well-known that they are described \nequivalently as (a) eigenstates of the usual annihilation operator, (b) from \na displacement operator acting on a fundamental state and (c) as minimum \nuncertainty states. What we observe in the different generalizations proposed \nis that the preceding definitions are no longer equivalent and only some of \nthe properties of the harmonic oscillator CS are preserved. \nIn this thesis we propose to study a new class of coherent states and \nits properties. We note that in one example our CS coincide with the ones \nproposed by Glauber where a set of three requirements for such states has \nbeen imposed. The set of our generalized coherent states remains invariant \nunder the corresponding time evolution and this property is called temporal \nstability. Secondly, there is no state which is orthogonal to all coherent states (the coherent states form a total set). The third property is that we \nget all coherent states by acting on one of these states [¿fiducial vector¿] with \noperators. They are highly non-classical states, in the sense that in general, \ntheir Bargmann functions have zeros which are related to negative regions of \ntheir Wigner functions. Examples of these coherent states with Bargmann \nfunction that involve the Gamma and also the Riemann ¿ functions are represented. \nThe zeros of these Bargmann functions and the paths of the zeros \nduring time evolution are also studied.
For a system with one degree of freedom, coherent states that are parametrized by classical canonical action-angle variables are introduced. These states also possess continuity of labelling, a resolution of unity, and temporal stability. The insistence on canonical action-angle variables strongly restricts any remaining arbitrariness in the coherent state definition. Such states are introduced for semibounded Hamiltonian operators having either a discrete or a continuous spectrum. Hamiltonians that have both discrete and continuous parts in their spectrum are also discussed.
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 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 ...