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Ibn AL- Haitham Journal For Pure and Applied Science

Determination of the Minimum Uncertainty States for a Rosen-Morse Potential by Using the Canonical Transformation of the Hamiltonian of the System

W.R. Mohsen

Published Oct 20, 2005• 0 citations
Publisher

Abstract

The wave functions of the minimum uncertainty states for a one- dimensional Rosen-Morse potential are obtained via exploiting Nieto's formalism for the construction of minimum uncertainty coherent states for different one-dimensional potentials and by using a canonical transformation of the Hamiltonian of the problem into a new Hamiltonian which is chosen so that it looks like a harmonic oscillator the mathematical derivations of the Nieto's procedures are presented.

DOI
10.30526/18.4.4402

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The wave functions of the minimum uncertainty states for a one- dimensional Rosen-Morse potential are obtained via exploiting Nieto's formalism for the construction of minimum uncertainty coherent states for different one-dimensional potentials and by using a canonical transformation of the Hamiltonian of the problem into a new Hamiltonian which is chosen so that it looks like a harmonic oscillator the mathematical derivations of the Nieto's procedures are presented.
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.
We obtain the exact, normalized, closed-form eigenfunctions for the one-dimensional Morse oscillator, as well as the raising and lowering operators. We next review a new method for obtaining the coherent states for arbitrary potentials, it being based on the classical motion of a particle. We apply this method to the Morse oscillator. After demonstrating that in the appropriate limit our results reduce to those for the harmonic oscillator, we obtain analytic insights into published numerical results on uncertainty products for Morse-oscillator wave packets.
The Morse potential one-dimensional quantum system is a realistic model for studying vibrations of atoms in a diatomic molecule. This system is very close to the harmonic oscillator one. We thus propose a construction of squeezed coherent states similar to the one of the harmonic oscillator using ladder operators. The properties of these states are analysed with respect to the localization in position, minimal Heisenberg uncertainty relation, the statistical properties and illustrated with examples using the finite number of states in a well-known diatomic molecule. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.
The coherent states of the Morse potential that have been obtained earlier from supersymmetric quantum mechanics, are shown to be connected with the representations of the affine group of the real line and some of its extensions. This relation is similar to the one between the Heisenberg-Weyl group and the coherent states of the harmonic oscillator. The states that minimize the uncertainty product of the generators of the affine Lie algebra are shown to contain all the coherent states of the Morse oscillator plus the intelligent states of the Morse Hamiltonians with different shape parameter s . The representations of the central extension of the affine group denoted by G 0 and its further extension will be shown to define the phase space relevant to the problem by choosing an appropriate orbit of the coadjoint representation of . This allows one to construct a generalized Wigner function on this phase space, which is again essentially in the same relation with the affine group, as the ordinary Wigner function with the Heisenberg-Weyl group.
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.
The minimum-uncertainty coherent-states formalism is extended to higher-dimensional systems. Specifically, for spherically symmetric three-dimensional potentials the formalism looks for coherent states which are products of an angular wave function times a radial wave function. After reviewing the many studies on angular coherent states, I concentrate on the physically distinguishing radial coherent states. The radial formalism is explained in detail and contrasted with the effective one-dimensional formalism. The natural classical variables in the radial formalism are those which vary sinusoidally as $g(E, L)\ensuremath{\theta}(t)$, where $\ensuremath{\theta}(t)$ is the real azimuthal angular variable and g(E,L)g(E, L)g(E,L) is the number of oscillations between apsidal distances per classical orbit. When changed to natural quantum operators, these operators can be given as the Hermitian sums and differences of the "lll" raising and lowering operators. The formalism is applied to the three-dimensional harmonic-oscillator and Coulomb problems.