Operators for the two-dimensional harmonic oscillator in an angular momentum basis

Abstract

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.