Abstract
The three-dimensional isotropic quantum harmonic oscillator (QHO) plays important roles in physics. Despite its importance, much of its rich mathematical structure and symmetry are often not sufficiently stressed, particularly at the advanced undergraduate or beginning graduate levels. In this paper, the well-known yet less apparent SU(3) symmetry of the three-dimensional isotropic QHO is used to yield its energy eigenvalues. In particular, the relatively less explored cubic Casimir operator of the SU(3) Lie group is exploited to obtain these well-established energy eigenvalues. The relationship between the degeneracy of each energy level of the three-dimensional QHO and the dimensions of irreducible representations of the SU(3) group is revisited and elaborated.
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