No data available.

Please log in to see this content.

You have no subscription access to this content.

No metrics data to plot.

The attempt to load metrics for this article has failed.

The attempt to plot a graph for these metrics has failed.

The full text of this article is not currently available.

f

Comments on Separability Operators, Invariance Ladder Operators, and Quantization of the Kepler Problem in Prolate‐Spheroidal Coordinates

### Abstract

The Schrödinger equation for the hydrogen atom separates in three coordinate systems: spherical, parabolic, and prolate spheroidal. The separability operators associated with the separation constants for these three systems are exhibited and discussed. Also, for these systems, the invariance ladder operators which transform a simultaneous eigenfunction of the separability operators into a different simultaneous eigenfunction of the same energy are discussed with reference to the elements of the *O* _{4}Lie algebra. Quantization of the Kepler problem in terms of prolate spheroidal coordinates is accomplished and discussed.

© 1968 The American Institute of Physics

Received 10 July 1967
Published online 28 October 2003

/content/aip/journal/jmp/9/4/10.1063/1.1664620

1.

1.For example, see M. Bander and C. Itzykson, Rev. Mod. Phys. 38, 330, 346 (1966) and references therein.

2.

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.

3.V. Bargmann, Z. Physik 99, 576 (1936).

4.

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.

5.Here we use the fact that

6.

6.Here the unit vector is taken in the 3‐direction.

7.

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.

8.The ladder operators given here have not been normalized to generate normalized wavefunctions from normalized wavefunctions.

9.

9.See for example, L. I. Schiff, Quantum Mechanics (McGraw‐Hill Book Company, Inc., New York, 1955), 2nd ed., p. 89.

11.

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.

12.Bateman Manuscript Project, Higher Transcendental Functions, A. Erdélyi, Ed. (McGraw‐Hill Book Company, Inc., New York, 1953), Vol. 2, p. 317.

13.

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.

14.See Eq. (40) and the discussion preceding it for a description of the roots α.

15.

15.In Fig. 1, only the resultant cuts in the z plane for rational functions of z and are shown.

16.

16.For an example employing this technique, see H. Goldstein, Classical Mechanics (Addison‐Wesley Publishing Co., Inc., Reading Mass., 1950), p. 302.

http://aip.metastore.ingenta.com/content/aip/journal/jmp/9/4/10.1063/1.1664620

Article metrics loading...

/content/aip/journal/jmp/9/4/10.1063/1.1664620

2003-10-28

2016-09-27

Full text loading...

###
Most read this month

Article

content/aip/journal/jmp

Journal

5

3

true

Commenting has been disabled for this content