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

Bracken A.J. and Leemon H.I. (1979) The isotropic harmonic oscillator in an angular momentum basis: An algebraic formulation. Journal of Mathematical Physics, 21 8: 2170-2181.

Author Bracken A.J.
Leemon H.I.
Title The isotropic harmonic oscillator in an angular momentum basis: An algebraic formulation
Journal name Journal of Mathematical Physics   Check publisher's open access policy
ISSN 0022-2488
Publication date 1979
Sub-type Article (original research)
Volume 21
Issue 8
Start page 2170
End page 2181
Total pages 12
Subject 1605 Policy and Administration
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 harnomic oscillator, and L is its angular momentum vector. It is shown that H can be written in the form ℏω(2ν†ν + 놕λ + 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 놕λ, which has nonnegative integer eigenvalues l, the total angular momentum quantum number. Thus the eigenvalues of H appear in the familiar form ℏω(2k + l + 3/2), previously obtained only by working in the coordinate or momentum representation. The common eigenvectors are constructed by applying 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.
Q-Index Code C1
Q-Index Status Provisional Code
Institutional Status Unknown

Document type: Journal Article
Sub-type: Article (original research)
Collection: Scopus Import
 
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Created: Tue, 26 Jul 2016, 05:08:56 EST by System User