Group Theory of Harmonic Oscillators in n-Dimensional Space

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Abstract
It is shown how the states of N particles which move in a common n‐dimensional harmonic oscillator potential can be classified according to the irreducible representations of the unitary groups UN or Un. The complete set of Nn independent integrals of the motion is obtained, their simultaneous eigenvectors being generalized Gel'fand basis vectors which can be enumerated either as bases for irreducible representations of UN or Un. Explicit formulas are given for the linear transformations induced on the basis vectors by the infinitesimal operators of UN and Un. The relation of the present work for n = 3 to that of Bargmann and Moshinsky is noted.