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Harmonic oscillator tensors. I. The nondegenerate case
Author(s) -
Palting Pancracio
Publication year - 1991
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560400403
Subject(s) - harmonic oscillator , hamiltonian (control theory) , basis (linear algebra) , quantum harmonic oscillator , lie algebra , mathematics , tensor (intrinsic definition) , quantum algebra , pure mathematics , tensor product , tensor operator , algebra over a field , mathematical physics , irreducible representation , quantum , quantum mechanics , physics , algebra representation , spherical harmonics , geometry , mathematical optimization
It is shown that the Heisenberg Lie algebra of the nondegenerate harmonic oscillator leads to a basis { J + , J 0 , J − } of LASU (2). The Hamiltonian of the system is proportional to J 0 , 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.