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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Coherent states for general potentials. V. Time evolution

Vincent P. Gutschick, Michael Martin Nieto

Published Jul 15, 1980• 76 citations
Publisher

Abstract

We show the time evolution of minimum-uncertainty coherent-state (MUCS) wave packets in the solvable potentials we have considered. (Numerical techniques are discussed in the appendices.) The time evolution is compared to the motion that a classical particle would have in the same potential. We make a number of observations on the conditions which can cause the states to lose their coherence more (or less) rapidly with time, and compare the MUCS results with those which can be obtained from other types of coherent states. The most physically interesting comparison is with the "continuous representation" coherent states.

DOI
10.1103/physrevd.22.403

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We apply our minimum-uncertainty coherent-states (MUCS) formalism to two one-dimensional systems that have continua: the symmetric Rosen-Morse potential and the Morse potential. The coherent states are discussed analytically in great detail, and the connections to annihilation-operator and displacement-operator coherent states are given. For the Rosen-Morse system the existence of a continuum does not prevent one from obtaining the coherent states in analytic, closed form. The Morse system, with its energy-dependent natural classical variable ${X}_{c}$, has a natural quantum operator $X$ which is Hamiltonian-dependent. This Hamiltonian dependence is complicated and prevents an easy analytic solution for the MUCS. However, approximate MUCS can be obtained by analytic approximation techniques.

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•Jan 1, 2000•2 citations
We show the time evolution of minimum-uncertainty coherent-state (MUCS) wave packets in the solvable potentials we have considered. (Numerical techniques are discussed in the appendices.) The time evolution is compared to the motion that a classical particle would have in the same potential. We make a number of observations on the conditions which can cause the states to lose their coherence more (or less) rapidly with time, and compare the MUCS results with those which can be obtained from other types of coherent states. The most physically interesting comparison is with the "continuous representation" coherent states.
We apply our minimum-uncertainty coherent-states (MUCS) formalism to two one-dimensional systems that have continua: the symmetric Rosen-Morse potential and the Morse potential. The coherent states are discussed analytically in great detail, and the connections to annihilation-operator and displacement-operator coherent states are given. For the Rosen-Morse system the existence of a continuum does not prevent one from obtaining the coherent states in analytic, closed form. The Morse system, with its energy-dependent natural classical variable Xc{X}_{c}Xc​, has a natural quantum operator XXX which is Hamiltonian-dependent. This Hamiltonian dependence is complicated and prevents an easy analytic solution for the MUCS. However, approximate MUCS can be obtained by analytic approximation techniques.
From the motivation of Schroedinger, that of finding states which follow the motion which a classical particle would have in a given potential, we discuss generalizations of the coherent states of the harmonic oscillator. We focus on a method which is the analytic complement to the group theory point of view. It uses a minimum uncertainty formalism as its basis. We discuss the properties and time evolution of these states, always keeping in mind the desire to find quantum states which follow the classical motion.
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.
We derive from a dynamical symmetry property that the linear and nonlinear Schr\"odinger equations with harmonic potential possess an infinite string of shape-preserving coherent wave-packet states with classical motion. Unlike the Schr\"odinger state with $\ensuremath{\Delta}x\ensuremath{\Delta}p=\frac{\ensuremath{\hbar}}{2}$, the uncertainty product can be arbitrarily large for these states showing that classical motion is not necessarily linked with minimum uncertainty. We obtain a generalization of Sudarshan's diagonal coherent-state representation in terms of these states.