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International Journal of Modern Physics B

EXACT SOLUTIONS, LADDER OPERATORS AND BARUT–GIRARDELLO COHERENT STATES FOR A HARMONIC OSCILLATOR PLUS AN INVERSE SQUARE POTENTIAL

Shi‐Hai Dong, Marcelo Lozada‐Cassou

Published Nov 10, 2005• 17 citations
Publisher

Abstract

We present exact solutions of the one-dimensional Schrödinger equation with a harmonic oscillator plus an inverse square potential. The ladder operators are constructed by the factorization method. We find that these operators satisfy the commutation relations of the generators of the dynamical group SU(1, 1). Based on those ladder operators, we obtain the analytical expressions of matrix elements for some related functions ρ and [Formula: see text] with ρ=x2. Finally, we make some comments on the Barut–Girardello coherent states and the hidden symmetry between E(x) and E(ix) by substituting x→ix.

DOI
10.1142/s0217979205032735

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We present exact solutions of the one-dimensional Schrödinger equation with a harmonic oscillator plus an inverse square potential. The ladder operators are constructed by the factorization method. We find that these operators satisfy the commutation relations of the generators of the dynamical group SU(1, 1). Based on those ladder operators, we obtain the analytical expressions of matrix elements for some related functions ρ and [Formula: see text] with ρ=x2. Finally, we make some comments on the Barut–Girardello coherent states and the hidden symmetry between E(x) and E(ix) by substituting x→ix.
The exact solutions of the Schrödinger equation with a harmonic oscillator plus an inverse square potential are obtained in two dimensions. We construct the ladder operators directly from the radial wave functions and find that these operators satisfy the commutation relations of an SU (1, 1) group. We obtain the explicit expressions of the matrix elements for some related functions ρ and [Formula: see text] with ρ = r2. We also explore another symmetry between the eigenvalues E(r) and E(ir) by substituting r→ir.
We present exact solutions of the Schrödinger equation with a harmonic oscillator plus an inverse squared potential in arbitrary dimensions. The ladder operators for the radial wavefunctions are established by the factorization method. We find that those operators together with satisfy the commutation relation of the generators of an su(1, 1) algebra. Some comments are made on the general calculation formula for off-diagonal matrix elements and the recurrence relation among them. Also, we show that this anharmonic oscillator possesses a hidden symmetry between E ( r ) and E (i r ) by substituting r → i r .
The eigenfunctions and eigenvalues of the three-dimensional Schrödinger equation with a harmonic oscillator plus an inverse square interaction are obtained. A realization of the ladder operators for the wave functions is studied. It is found that these operators satisfy the commutation relations of an SU(1,1) group. The closed analytical expressions for the matrix elements of different functions ρ and ρd/dρ with ρ = r 2 are evaluated. Another hidden symmetry explores the relations between the eigenvalues and eigenfunctions for substituting r→ ir. PACS number(s): 03. 65. Fd, 03. 65.Ge and 02. 20.Qs.
Abstract The exact solutions of a one‐dimensional Schrödinger equation with a harmonic oscillator plus an inverse square potential are obtained. The ladder operators constructed directly from the normalized wavefunctions are found to satisfy a su(1, 1) algebra. Another hidden symmetry is used to explore the relations between the eigenvalues and eigenfunctions by substituting x → −ix. The vibrational partition function Z is calculated exactly to study thermodynamic functions such as the vibrational mean energy U, specific heat C, free energy F, and entropy S. It is both interesting and surprising to find that both vibrational specific heat C and entropy S are independent of the potential strength α. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007
In this paper we try to introduce the ladder operators associated with the pseudoharmonic oscillator, after solving the corresponding Schrödinger equation by using the factorization method. The obtained generalized raising and lowering operators naturally lead us to the Dirac representation space of the system which is much easier to work with, in comparison to the functional Hilbert space. The SU (1, 1) dynamical symmetry group associated with the considered system is exactly established through investigating the fact that the deduced operators satisfy appropriate commutation relations. This result enables us to construct two important and distinct classes of Barut—Girardello and Gilmore—Perelomov coherent states associated with the system. Finally, their identities as the most important task are exactly resolved and some of their nonclassical properties are illustrated, numerically.
In this paper we try to introduce the ladder operators associated with the pseudoharmonic oscillator, after solving the corresponding Schrödinger equation by using the factorization method. The obtained generalized raising and lowering operators naturally lead us to the Dirac representation space of the system which is very easier to work with, in comparison to the functional Hilbert space. The SU(1,1) dynamical symmetry group associated with the considered system is exactly established through investigating the fact that the deduced operators satisfy appropriate commutation relations. This result enables us to construct two important and distinct classes of Barut-Girardello and Gilmore-Perelomov coherent states associated with the system. Finally, their identities as the most important task are exactly resolved and some of their nonclassical properties are illustrated, numerically.