Degeneracies of the spherical well, harmonic oscillator and hydrogen atom in arbitrary dimensions

American Journal of Physics

Volumn 64, Issue 4, 1996, Pages 430-434

  Shea, Richard W ^a   Aravind, P K ^a  

^a WORCESTER POLYTECHNIC INSTITUTE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030528640     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.18185     Document Type: Article

References (21)

1. Some books that explore group theoretical aspects of the oscillator or hydrogen atom are: B. G. Wybourne, Classical Groups for Physicists (Wiley, New York, 1974); M. J. Engelfield, Group Theory and the Coulomb Problem (Wiley-Interscience, New York, 1972); O. L. deLange and R. E. Raab, Operator Methods in Quantum Mechanics (Oxford U.P., New York, 1991).
2. Some books that explore group theoretical aspects of the oscillator or hydrogen atom are: B. G. Wybourne, Classical Groups for Physicists (Wiley, New York, 1974); M. J. Engelfield, Group Theory and the Coulomb Problem (Wiley-Interscience, New York, 1972); O. L. deLange and R. E. Raab, Operator Methods in Quantum Mechanics (Oxford U.P., New York, 1991).
3. Some books that explore group theoretical aspects of the oscillator or hydrogen atom are: B. G. Wybourne, Classical Groups for Physicists (Wiley, New York, 1974); M. J. Engelfield, Group Theory and the Coulomb Problem (Wiley-Interscience, New York, 1972); O. L. deLange and R. E. Raab, Operator Methods in Quantum Mechanics (Oxford U.P., New York, 1991).
4. G. A. Baker, "Degeneracy of the n-dimensional, isotropic harmonic oscillator," Phys. Rev. 103, 1119-1120 (1956).
5. M. Bander and C. Itzykson, "Group theory and the hydrogen atom (I) and (II)," Rev. Mod. Phys. 38, 330-345 (1966); 38, 346-358 (1966).
6. M. Bander and C. Itzykson, "Group theory and the hydrogen atom (I) and (II)," Rev. Mod. Phys. 38, 330-345 (1966); 38, 346-358 (1966).
7. M. M. Nieto, "Hydrogen atom and relativistic pi-mesic atom in N-space dimensions," Am. J. Phys. 47, 1067-1072 (1979).
8. P. J. Yanez, W. Van Assche, and J. S. Dehesa, "Position and momentum information entropies of the D-dimensional harmonic oscillator and hydrogen atom," Phys. Rev. A 50, 3065-3079 (1994).
9. Some tutorial articles on supersymmetry are A. R. P. Rau and R. W. Haymaker, "Supersymmetry in quantum mechanics," Am. J. Phys. 54, 928-936 (1986); R. Dutt, A. Khare, and U. P. Sukhatme, "Supersymmetry, shape invariance and exactly solvable potentials," Am. J. Phys. 56, 163- 168 (1988).
10. Some tutorial articles on supersymmetry are A. R. P. Rau and R. W. Haymaker, "Supersymmetry in quantum mechanics," Am. J. Phys. 54, 928- 936 (1986); R. Dutt, A. Khare, and U. P. Sukhatme, "Supersymmetry, shape invariance and exactly solvable potentials," Am. J. Phys. 56, 163-168 (1988).
11. V. A. Kostelecky and M. M. Nieto, "Evidence for a phenomenological supersymmetry in atomic physics," Phys. Rev. Lett. 53, 2285-2288 (1984).
12. V. A. Kostelecky, M. M. Nieto, and D. R. Traux, "Supersymmetry and the relation between the Coulomb and oscillator problems in arbitrary dimensions," Phys. Rev. D 32, 2627-2633 (1985).
13. M. Kibler, A. Ronveaux, and T. Negadi, "On the hydrogen-oscillator connection: passage formulas between wavefunctions," J. Math. Phys. 27, 1541-1548 (1986).
14. G. Zeng, K. So, and M. Li, "Most general and simplest algebraic relationship between energy cigenstates of a hydrogen atom and a harmonic oscillator of arbitrary dimensions," Phys. Rev. A 50, 4373-4375 (1994).
15. D. S. Bateman, C. Boyd, and B. Dutta-Roy, "The mapping of the Coulomb problem into the oscillator," Am. J. Phys. 60, 833-836 (1992).
16. d-2), one finds that the total number is given by Eq. (5) of the text.
17. A. Erdelyi, Bateman Manuscript Project: Higher Transcendental Functions, Vol. II (McGraw-Hill, New York, 1953-1955), Chap. XI.
18. i are non-negative integers that add up to n. The points (n,0,...,0), (0,n,...0),...(0,0,...,n) define the vertices of a (d - 1)-dimensional regular simplex within which all the other points lie. The entire set of points is invariant under the d! permutations of their coordinates, which constitute all the symmetry operations of a (d - 1)-dimensional regular simplex, and hence ail the points must be disposed symmetrically within such a solid. Finally the distance between neighboring points of the set is √2, which is larger than the unit diameter of the spheres centered at the points.
19. J. H. Van Vleck, in Wave Mechanics, The First Fifty Years, edited by W. C. Price et al. (Butterworths, London, 1973), pp. 26-37.
20. D. Z. Goodson, D. K. Watson, J. G. Loeser, and D. R. Herschbach. "Energies of doubly excited two-electron atoms from interdimensional degeneracies," Phys. Rev. A 44, 97-102 (1991).
21. By the parity operation we mean a reversal in the signs of all the spatial coordinates of an arbitrary point. In an even number of dimensions our parity operation is equivalent to a rotation but in an odd number of dimensions it is not.