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arXiv:quant-ph/0302171

Coherent States: A General Approach

P. K. Panigrahi, T. Shreecharan, J. Banerji, Sundaram, V.

Published Feb 22, 2003• 0 citations
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Abstract

A general procedure for constructing coherent states, which are eigenstates of annihilation operators, related to quantum mechanical potential problems, is presented. These coherent states, by construction are not potential specific and rely on the properties of the orthogonal polynomials, for their derivation. The information about a given quantum mechanical potential enters into these states, through the orthogonal polynomials associated with it and also through its ground state wave function. The time evolution of some of these states exhibit fractional revivals, having relevance to the factorization problem.

DOI
10.48550/arxiv.quant-ph/0302171

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A general procedure for constructing coherent states, which are eigenstates of annihilation operators, related to quantum mechanical potential problems, is presented. These coherent states, by construction are not potential specific and rely on the properties of the orthogonal polynomials, for their derivation. The information about a given quantum mechanical potential enters into these states, through the orthogonal polynomials associated with it and also through its ground state wave function. The time evolution of some of these states exhibit fractional revivals, having relevance to the factorization problem.
A general algebraic procedure for constructing coherent states of a wide class of exactly solvable potentials, e.g., Morse and P\"oschl-Teller, is given. The method, a priori, is potential independent and connects with earlier developed ones, including the oscillator-based approaches for coherent states and their generalizations. This approach can be straightforwardly extended to construct more general coherent states for the quantum-mechanical potential problems, such as the nonlinear coherent states for the oscillators. The time evolution properties of some of these coherent states show revival and fractional revival, as manifested in the autocorrelation functions, as well as, in the quantum carpet structures.
We first review the properties of the harmonic-oscillator coherent states which can be equivalently defined as (a) a specific subset of the $x\ensuremath{-}p$ minimum-uncertainty states, (b) eigenstates of the annihilation operator, or (c) states created by a certain unitary exponential displacement operator. Then we present a new method for finding coherent states for particles in general potentials. Its basis is the desire to find those states which most nearly follow the classical motion, but it is most nearly a generalization of the minimum-uncertainty method. The properties of these states are discussed in detail. Next we show that the annihilation operator and displacement operator methods, as heretofore defined, cannot be applied to general potentials (whose eigenvalues are not equally spaced). We define a generalization of these methods but show that the states so defined are not, in general, equivalent to the minimum-uncertainty coherent states. We discuss a number of properties of our coherent states and the procedures we have used.
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.
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.
We introduce a right inverse for the annihilation operator which appears in the factorization of associated hypergeometric-type operators. This makes possible the definition of generalized coherent states for the associated eigenfunctions. These generalized coherent states satisfy the relevant properties, i.e., label continuity, overcompleteness, temporal stability and action identity required to establish a close connection between the quantum and classical formulations of a given physical system.