Factorizations of vector operators for the isotropic harmonic oscillator in an angular momentum basis

Abstract

The factorization of four vector operators, D±(ω) and D±(−ω), which occur in a representation-independent, spectrum-generating algebra for the three-dimensional, isotropic harmonic oscillator in an angular momentum basis, is considered (ω is the angular frequency of the oscillator). The D±(ω) are quantum-mechanical analogs of the classical vectors (1∓iL̂×)Fc (ω), where Fc(ω)=−Mωr×L+pL is constant in a frame rotating with angular velocity ωL̂. It is shown that these four vector operators can be factorized in two different ways to yield operators that, apart from their dependence on a constant of the motion (L2), are linear in either p or r. In this way 20 abstract operators are obtained. The properties of these operators are discussed: (i) Twelve are ladder operators for the quantum numbers l, and l and m, in the eigenkets ‖lm〉 of L2 and Lz. In linearized, differential form six of these operators are ladder operators for the spherical harmonics in the coordinate representation, while the other six are the corresponding operators in the momentum representation. (ii) The remaining eight operators factorize linear combinations of the Hamiltonian and the dimension operator. In linearized, differential form four of these operators are ladder operators for energy and angular momentum in the radial part of the coordinate-space wave functions, while the other four are the corresponding operators in the momentum representation.