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Journal of Physics A Mathematical and General

The correspondence of two definitions of coherent states in a particular system

M. G. A. Crawford

Published Jan 1, 1999• 1 citation
Publisher

Abstract

Many generalizations of the harmonic oscillator coherent states are known in the literature. One such generalization by Nieto has been applied to a particular system known as the `harmonic oscillator with centripetal barrier'. In this paper, it is shown that Nieto's generalization in this system may be described in terms of a recent generalization due to Klauder. This is of considerable interest since the approaches to generalizing coherent states differ significantly.

DOI
10.1088/0305-4470/32/19/101

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Many generalizations of the harmonic oscillator coherent states are known in the literature. One such generalization by Nieto has been applied to a particular system known as the `harmonic oscillator with centripetal barrier'. In this paper, it is shown that Nieto's generalization in this system may be described in terms of a recent generalization due to Klauder. This is of considerable interest since the approaches to generalizing coherent states differ significantly.
Coherent states in the harmonic oscillator have long been thought of as a bridge from quantum mechanics to classical mechanics. Many of their important properties (most notably those involving time evolution) are only valid in the harmonic oscillator and are at best approximations elsewhere. This thesis is an investigation into several means of generalizing coherent states for other systems and examines the resultant states for some systems. One system of interest is the harmonic oscillator with centripetal barrier for which coherent states of several disparate definitions coincide. Also studied is this spherical rotator, a system which is particularly amenable to defining annihilation operator coherent states. A third system under investigation is the hydrogen atom. This system serves as an arena for the development of an extension to a generalization due to Klauder [J. Phys. A. 29(12):L293-L298. 1996]. Klauder's construction is only applicable to systems without energy degeneracies and must be extended for application where degeneracies are present. The author provides a means for this extension and applies the complete construction to the hydrogen atom problem. As a demonstration of how this construction may be adapted, the author constructs Rydberg wave packets which are initially localized and exhibit full and fractional revivals in the long time evolution.
Coherent states in the harmonic oscillator may be defined in several equivalent ways. One definition describes coherent states as special states satisfying a minimum-uncertainty requirement in position and momentum spaces. This definition is generalized for other potentials according to a method developed by Nieto et al. [Phys. Rev. D 20, 1321 (1979)] and is herein applied to the P\"oschl-Teller potential. A classical limit based on the harmonic-oscillator coherent-state classical limit is then developed and applied to the P\"oschl-Teller minimum-uncertainty states. In this limit, classical behavior may be obtained from these quantum states. Together with a completeness argument for the generalized coherent states, this result provides insight into quantum classical correspondence through the statistical interpretation of quantum mechanics.
We apply our minimum-uncertainty coherent-states formalism, which is physically motivated by the classical motion, to two confining one-dimensional systems: the harmonic oscillator with centripetal barrier and the symmetric P\"oschl-Teller potentials. The minimum-uncertainty coherent states are discussed in great detail, and the connections to annihilation-operator coherent states and displacement-operator coherent states are given. The first system discussed provides an excellent bridge between the harmonic oscillator and more general potentials because, even though it is a nonharmonic potential, its energy eigenvalues are equally spaced. Thus, its coherent states have many, but not all, of the properties of the harmonic-oscillator coherent states.
Gaussian Klauder coherent states are constructed for the harmonic oscillator, the planar rotor, and the particle in a box. The standard harmonic oscillator coherent states are given by expansions in the eigenstates of the Hamiltonian in terms of a complex parameter \ensuremath{\alpha}. When the complex modulus of \ensuremath{\alpha} is large, these states are identical in behavior with a particular choice of Gaussian Klauder coherent state. When the angular momentum of a planar rotor is large compared with Planck's constant, the angle distribution associated with a Gaussian Klauder coherent state for this case remains sharply localized for many rotations. Similarly, for the particle in a box, it is possible to choose parameters in the Gaussian Klauder coherent state so that a localized particle bounces back and forth at constant velocity between the walls of the box for many periods without significant delocalization. Buried in this behavior is the Fourier series for a triangle wave. These examples show how Gaussian Klauder coherent states are of utility in understanding quantum-classical correspondence.
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.