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Applications

Owen L. de Lange, R. E. Raab

Published Nov 28, 1991• 0 citations
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Abstract

Abstract We consider some applications of results obtained in the previous chapter. In Sections 6.1–6.3 operator methods are used to calculate matrix elements for the three-dimensional isotropic harmonic oscillator and the Coulomb potential. The shift operators also provide a simple method of constructing normalized coordinate space wavefunctions and some examples are given in Section 6.4. In Sections 6.5–6.8 we discuss applications to constant and sectionally constant potentials.

DOI
10.1093/oso/9780198539612.003.0006

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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.

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