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The semiclassical treatment of the two-dimensional harmonic oscillator provides an interesting example for the relation between classical motion and quantum mechanical energy spectrum. We extend previous work on the anisotropic oscillator with incommensurate frequencies [1] and the isotropic oscillator [2] to the case with commensurate frequencies [3]. There, Lissajous curves appear as additional classical periodic orbits. With three different scenarios depending on the
ratio of its frequencies, the two-dimensional harmonic oscillator offers a unique possibility to explicitly analyze the role of symmetries in classical and quantum mechanics.
[1] M. Brack and S. R. Jain, Phys. Rev. A 51, 3462 (1995) [2] S. C. Creagh, Ann. Phys. (N.Y.) 248, 60 (1996) [3] R. Doll and G.-L. Ingold, Am. J. Phys. 75, 208 (2007) |
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