Vol. 19, No. 1, 1966

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On the degeneracy of the Kepler problem

Victor A. Dulock and Harold V. McIntosh

Vol. 19 (1966), No. 1, 39–55

In an earlier paper we have shown a method which may be used to construct an SU(n) group as the symmetry group of the harmonic oscillator in classical mechanics. The method is applicable to quadratic Hamiltonians, and was applied in a subsequent paper to the charged harmonic oscillator in a magnetic field. We now apply the technique to the Kepler problem, which may be made equivalent to a harmonic oscillator by a suitable transformation. An SU(3) group is found, generated by constants of the motion which are the angular momentum, the Runge vector, which points to the perihelion, and a vector along the line of nodes. Different groups are found by separation in spherical and parabolic coordinates, while yet another group is found by inspection in parabolic coordinates. One purpose of our investigation is to find symmetry groups for quantum mechanical problems. While our results dispel the thought that there might not be sets of constants of the motion closed with respect to Poisson Brackets and thus generating a Lie symmetry group, they do show that the functional relationship involved may make it very difficult to use the correspondence principle to construct satisfactory quantum mechanical operators. Our SU(3) symmetry group is not isomorphic to the R(4) symmetry group of the hydrogen atom found by Fock and Bargmann; the angular momentum and Runge vector are nonlinear functions of a subset of the generators of our SU(3) group. It is not possible to find an operator generalization of these functions which can be satisfied by irreducible representations of SU(3) and R(4).

Mathematical Subject Classification
Primary: 70.22
Received: 17 September 1965
Published: 1 October 1966
Victor A. Dulock
Harold V. McIntosh