The classical Lenz vector and the two-dimensional quantum harmonic oscillator

Both the two-dimensional harmonic oscillator and the Newton potential allow particular solutions for the orbits which are ellipses with center of attraction in the center, in the !rst case, and in one focus, in the second. The same complex map which allows to go from Kepler’s to Hooke’s orbits, and back, is used to transform the Lenz vector, de!ned for the Kepler orbit, into two conserved quantities for the harmonic motion. Upon quantization, the resulting operators, together with the angular momentum Lz , are found to correspond to the generators of the SU(2) internal symmetry of the two-dimensional quantum oscillator and the connection to the Schwinger model of angular momentum is made apparent. We give a self-contained new look on this topic.