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Journal of Physics A Mathematical and General

An integrated approach to ladder and shift operators for the Morse oscillator, radial Coulomb and radial oscillator potentials

Ian L. Cooper

Published Apr 7, 1993• 79 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/26/7/018

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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.
The bound-state solutions and the su(1,1) description of the d-dimensional radial harmonic oscillator, the Morse, and the D-dimensional radial Coulomb Schrödinger equations are reviewed in a unified way using the point canonical transformation method. It is established that the spectrum generating su(1,1) algebra for the first problem is converted into a potential algebra for the remaining two. This analysis is then extended to Schrödinger equations containing some position-dependent mass. The deformed su(1,1) construction recently achieved for a d-dimensional radial harmonic oscillator is easily extended to the Morse and Coulomb potentials. In the last two cases, the equivalence between the resulting deformed su(1,1) potential algebra approach and a previous deformed shape invariance one generalizes to a position-dependent mass background a well-known relationship in the context of constant mass.
By invoking a standard variable transformation x → y ≡ x 2 that connects the eigenvalue problems of the hydrogen atom system (Coulomb potential) and the Kratzer oscillator system , we obtain a system of ladder operators Ω nl ± ( l = 0,1,2, ... , n − 1) that connect the degenerate energy eigenstates (belonging to different l -values) of the hydrogen atom system. Using such quantum number specific operators as the precursors, we develop a general method of constructing quantum number independent ladder operators ; and then investigate the general conditions for these operators to form a realization of the su (2) algebra. In the process, we demonstrate that, in general, a finite-dimensional linear vector space carries a realization of the su (2) algebra. Applying the method to the case at hand, we show that for the hydrogen atom system the operators and form an su (2) algebra. The Hamiltonian is identified with the Casimir operator of the algebra, thereby revealing a dynamical SU (2) group symmetry of the hydrogen atom system.
Abstract In Chapter 5 we considered the factorization of a radial Hamiltonian for a particle in a spherically symmetric potential which is independent of angular momentum. This factorization provides shift operators for energy and angular momentum in the eigenkets of the radial Hamiltonian for the oscillator and Coulomb potentials. These operators are linear functions of the momentum operator *p and they are also nonlinear functions of the position operator r: in the coordinate representation of wave mechanics they yield first-order differential operators. In this chapter we show that for the three-dimensional isotropic harmonic oscillator and the Coulomb potential, one can also construct shift operators for energy and angular momentum that are linear functions* of r and nonlinear functions of p. In the momentum representation of wave mechanics the shift operators yield first-order differential operators. Applications to the calculation of energy eigenvalues, matrix elements, and radial momentum-space wavefunctions are considered.