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Physica Scripta

Algebraic approach to a harmonic oscillator plus an inverse squared potential and its recurrence relation

Shishan Dong, Shi‐Hai Dong, Marcelo Lozada‐Cassou

Published Jan 11, 2006• 6 citations
Publisher

Abstract

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 .

DOI
10.1088/0031-8949/73/2/008

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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 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 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 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
The Morse oscillator, radial Coulomb and radial harmonic oscillator problems can be solved exactly using a variety of algebraic methods. These problems correspond to different realizations of the so(2,1) algebra and a comparison of the generators of the algebra may be used to identify mappings between each pair of systems. The resultant transition operators act as ladder, or energy changing, operators in the cases of the Coulomb and harmonic oscillator potentials, whereas they act as shift operators, acting at constant energy, in the case of the Morse potential. This is a consequence of the so(2,1) dynamical symmetry, whereby the Morse Hamiltonian is expressible solely in terms of the Casimir operator of the algebra. An alternative algebraic approach, the use of the method of supersymmetric quantum mechanics, or factorization, produces in each case a set of shift operators. Relations between the various ladder and shift operators may be identified by means of the appropriate mappings, and these results can be generalized so as to relate the one dimensional Morse oscillator to the radial Coulomb and radial harmonic oscillator potentials involving an arbitrary number of angular dimensions.