The isotropic harmonic oscillator in an angular momemtum basis: An algebraic formulation

Abstract

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.