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Journal of Mathematical Physics

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

Owen L. de Lange

Published Nov 1, 1987• 2 citations
Publisher

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.

DOI
10.1063/1.527757

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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.
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.
A completely algebraic and representation-independent solution is presented of the simultaneous eigenvalue problem for H, L2, and L3, where H is the Hamiltonian operator for the three-dimensional, isotropic harmonic oscillator, and L is its angular momemtum vector. It is shown that H can be written in the form h/ω(2ν°ν+λ°⋅λ+3/2), where ν° and ν are raising and lowering (boson) operators for ν°ν, which has nonnegative integer eigenvalues k; and λ° and λ are raising and lowering operators for λ°⋅λ, which has nonnegative integer eigenvalues l, the total angular momentum quantum number. Thus the eigenvalues of H appear in the familiar form h/ω(2k+l+3/2), previously obtained only by working in the coordinating the operators ν° and λ° to a ’’vacuum’’ vector on which ν and λ vanish. The Lie algebra so(2,1) ⊕ so(3,2) is shown to be a spectrum-generating algebra for this problem. It is suggested that coherent angular momentum states can be defined for the oscillator, as the eigenvectors of the lowering operators ν and λ. A brief discussion is given of the classical counterparts of ν,ν°, λ, and λ°, in order to clarify their physical interpretation.
Abstract In Chapter 4 we considered a ‘twodimensional’ problem in which L2 and Lz constitute a c.s.c.o. and we derived shift operators for the angular momentum quantum numbers l and m in the eigenkets |lm> of these observables. In Chapter 5 we discussed a ‘one-dimensional’ problem, namely the eigenvalue equation for the radial Hamiltonian (eqn (5.25)). Here we use results from Chapters 4 and 5 to construct shift operators for the eigenkets of the three dimensional problem (5.11)–(5.13).
The forms of the operators ν°, ν, λ°, λ, which enable one to write the Hamiltonian of the two-dimensional isotropic harmonic oscillator in the form H=ℏω(2ν°ν+λ°⋅λ+1), are presented. Here ν° and ν are, respectively, the raising and lowering operators for ν°ν, the ‘‘radial’’ quantum number operator, while λ° and λ are, respectively, the raising and lowering operators for M, the magnitude of the angular momentum operator. Corresponding to this decomposition of H in the angular momentum basis are the energy eigenvalues Ekm=ℏω(2k+‖m‖+1) with k=0, 1, 2, ⋅⋅⋅ and m=0, ±1, ±2, ⋅⋅⋅. Here k is a ‘‘radial’’ quantum number, and m is a ‘‘magnetic’’ quantum number. The commutation relations satisfied by the operators ν°, ν, λ°, and λ are also presented.
We show that the supersymmetric radial ladder operators of the three-dimensional isotropic harmonic oscillator are contained in the spherical components of the creation and annihilation operators of the system. Also, we show that the constants of motion of the problem, written in terms of these spherical components, lead us to second-order radial operators. Further, we show that these operators change the orbital angular momentum quantum number by two units and are equal to those obtained by the Infeld–Hull factorization method.
An algebraic (representation-independent) analysis is presented for the Dirac oscillator in an angular momentum basis. The analysis is based on shift operators for energy and angular momentum, and it is similar to that for a non-relativistic isotropic harmonic oscillator. The shift operators generate all the eigenkets of the Dirac oscillator from a 'vacuum' ket. The shift operations yield energy eigenvalues and certain matrix elements. The relationship to the factorization method is discussed.