Abstract
A dynamical group is constructed for the isotropic three-dimensional harmonic oscillator by forming the semi-direct product group
W
(3)⊗
Sp
(6,
R
), where
W
(3) is the Weyl group and
Sp
(6,
R
) the real symplectic group. A single representation of
W
(3)⊗
Sp
(6,
R
) is constructed, using the usual harmonic oscillator annihilation and creation operators for
W
(3) and their anticommutators for
Sp
(6,
R
), which can be spanned by the complete set of harmonic oscillator states. The group
W
(3)⊗
Sp
(6,
R
) is found to simplify the calculation of matrix elements of operators acting on harmonic oscillator states and to have a rich and useful subgroup structure.
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