Harmonic oscillator tensors. I. The nondegenerate case

It is shown that the Heisenberg Lie algebra of the nondegenerate harmonic oscillator leads to a basis {J₊, J₀, J_?} of LASU (2). The Hamiltonian of the system is proportional to J₀, and the basis elements give rise to irreducible tensors in the associative enveloping algebra of the Heisenberg Lie algebra. The construction of these irreducible tensors is studied with special attention being paid to the case in which they act upon a single vector space spanned by the harmonic oscillator basis functions. A tensor coupling rule is developed, and useful application is made of it in the calculation of general expressions for vibrational operators and their matrix elements. Throughout, the value of the additional algebraic quantum numbers (l, m) is emphasized.