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Canadian Journal of Physics

The vector-ladder operators for the rotation group O3

J. E. Hardy

Published Mar 1, 1993• 0 citations
Publisher

Abstract

The two vector-ladder operators, which step from any state [Formula: see text] in an irreducible multiplet in the space of product states of two commuting angular momenta, are defined and all their nonvanishing matrix elements are given, facilitating direct, straightforward calculation of the six nearby nonsibling states [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text].

DOI
10.1139/p93-023

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The two vector-ladder operators, which step from any state [Formula: see text] in an irreducible multiplet in the space of product states of two commuting angular momenta, are defined and all their nonvanishing matrix elements are given, facilitating direct, straightforward calculation of the six nearby nonsibling states [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text].
The exact solutions of the Schrödinger equation with a harmonic oscillator plus an inverse square potential are obtained in two dimensions. We construct the ladder operators directly from the radial wave functions and find that these operators satisfy the commutation relations of an SU (1, 1) group. We obtain the explicit expressions of the matrix elements for some related functions ρ and [Formula: see text] with ρ = r2. We also explore another symmetry between the eigenvalues E(r) and E(ir) by substituting r→ir.
Formulas are obtained which give the matrix representation, relative to a product basis, of the projection operator for the total angular momentum of a system. If the individual angular momenta are not too large, the matrix elements depend upon a small number of parameters, independent of the number of angular momenta coupled. Recurrence relations between elements and symmetry properties of elements are derived. These results enable one to perform the vector coupling of a large number of angular momenta in a relatively simple fashion. The connection between the matrix elements and vector coupling coefficients is discussed. Several important special cases are treated in detail.