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Comments on Separability Operators, Invariance Ladder Operators, and Quantization of the Kepler Problem in Prolate‐Spheroidal Coordinates
1.For example, see M. Bander and C. Itzykson, Rev. Mod. Phys. 38, 330, 346 (1966) and references therein.
2.E. C. G. Sudarshan, N. Mukunda, and L. O’Raifeartaigh, Phys. Letters 19, 322 (1963);
2.H. Bacry, Nuovo Cimento 41A, 222 (1966);
2.M. Y. Han, Nuovo Cimento 42B, 367 (1966);
2.R. H. Pratt and T. F. Jordan, Phys. Rev. 148, 1276 (1966);
2.R. Musto, Phys. Rev. 148, 1274 (1966).
3.V. Bargmann, Z. Physik 99, 576 (1936).
4.C. Runge, Vektoranalysis (B. G. Teubner, Leipzig, 1919), Vol. 1, p. 70;
4.W. Lenz, Z. Physik 24, 197 (1923);
4.see also W. Pauli, Z. Physik 36, 336 (1926);
4.V. Fock, Z. Physik 98, 145 (1935).
5.Here we use the fact that
6.Here the unit vector is taken in the 3‐direction.
7.The operator may be obtained most simply through a consideration of a linear combination of the elements of the Lie algebra that commute with It may also be obtained as a product of operators derived by the factorization method [see L. Infeld and T. E. Hull, Rev. Mod. Phys. 23, 21 (1951)], or for by a simplification of the expression
8.The ladder operators given here have not been normalized to generate normalized wavefunctions from normalized wavefunctions.
9.See for example, L. I. Schiff, Quantum Mechanics (McGraw‐Hill Book Company, Inc., New York, 1955), 2nd ed., p. 89.
11.The variables ξ and η obtain the values only for the special case In the present analysis, we consider only the situations for which
12.Bateman Manuscript Project, Higher Transcendental Functions, A. Erdélyi, Ed. (McGraw‐Hill Book Company, Inc., New York, 1953), Vol. 2, p. 317.
13.Similar analytical difficulties are associated with the solutions of the separated equations, Eqs. (23) and (24). Even for the free particle the solutions involve Lame or spheroidal wave‐functions. [See, for example, W. Magnus and F. Oberhettinger, Formulas and Theorems for the Functions of Mathematical Physics (Chelsea Publishing Company, New York, 1949), p. 158;
13.C. Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford California, 1957)].
14.See Eq. (40) and the discussion preceding it for a description of the roots α.
15.In Fig. 1, only the resultant cuts in the z plane for rational functions of z and are shown.
16.For an example employing this technique, see H. Goldstein, Classical Mechanics (Addison‐Wesley Publishing Co., Inc., Reading Mass., 1950), p. 302.
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