Accidental degeneracy in a simple quantum system: A new symmetry group for a particle in an impenetrable square-well potential

American Journal of Physics

Volumn 65, Issue 11, 1997, Pages 1087-1094

  Leyvraz, F ^a   Frank, A ^a,b   Lemus, R ^b   Andres M V ^c  

^a INSTITUTO DE CIENCIAS FÍSICAS   (Mexico)

^b UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO   (Mexico)

^c UNIVERSITY OF SEVILLE   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031494203     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.18734     Document Type: Article

References (20)

1. See, e.g., J. M. Jauch and E. L. Hill, "On the problem of degeneracy in quantum mechanics," Phys. Rev. 57, 641-645 (1940); H. V. McIntosh, "Symmetry and Degeneracy," in Group Theoretical Methods and its applications, edited by E. M. Loebl (Academic, New York, 1990), Vol. II, pp. 75-137; M. Moshinsky, "Accidental Degeneracies and Symmetry Groups," Found. Phys. 13, 73-79 (1983).
2. See, e.g., J. M. Jauch and E. L. Hill, "On the problem of degeneracy in quantum mechanics," Phys. Rev. 57, 641-645 (1940); H. V. McIntosh, "Symmetry and Degeneracy," in Group Theoretical Methods and its applications, edited by E. M. Loebl (Academic, New York, 1990), Vol. II, pp. 75-137; M. Moshinsky, "Accidental Degeneracies and Symmetry Groups," Found. Phys. 13, 73-79 (1983).
3. See, e.g., J. M. Jauch and E. L. Hill, "On the problem of degeneracy in quantum mechanics," Phys. Rev. 57, 641-645 (1940); H. V. McIntosh, "Symmetry and Degeneracy," in Group Theoretical Methods and its applications, edited by E. M. Loebl (Academic, New York, 1990), Vol. II, pp. 75-137; M. Moshinsky, "Accidental Degeneracies and Symmetry Groups," Found. Phys. 13, 73-79 (1983).
4. L. Hulthen, "Über die Quanten Mechanische Herleitung der Balmerterme," Z. Phys. 86, 21-23 (1933); V. Fock, "Zur Theorie des Wasserstoffatoms," ibid. 98, 145-154 (1936); M. J. Englefield, Group Theory and the Coulomb Problem (Wiley Interscience, New York, 1972); X. L. Yang, M. Lieber, and F. T. Chan, "The Runge Lenz Vector for the Two-dimensional Hydrogen Atom," Am. J. Phys. 59, 231-232 (1991).
5. L. Hulthen, "Über die Quanten Mechanische Herleitung der Balmerterme," Z. Phys. 86, 21-23 (1933); V. Fock, "Zur Theorie des Wasserstoffatoms," ibid. 98, 145-154 (1936); M. J. Englefield, Group Theory and the Coulomb Problem (Wiley Interscience, New York, 1972); X. L. Yang, M. Lieber, and F. T. Chan, "The Runge Lenz Vector for the Two-dimensional Hydrogen Atom," Am. J. Phys. 59, 231-232 (1991).
6. L. Hulthen, "Über die Quanten Mechanische Herleitung der Balmerterme," Z. Phys. 86, 21-23 (1933); V. Fock, "Zur Theorie des Wasserstoffatoms," ibid. 98, 145-154 (1936); M. J. Englefield, Group Theory and the Coulomb Problem (Wiley Interscience, New York, 1972); X. L. Yang, M. Lieber, and F. T. Chan, "The Runge Lenz Vector for the Two-dimensional Hydrogen Atom," Am. J. Phys. 59, 231-232 (1991).
7. L. Hulthen, "Über die Quanten Mechanische Herleitung der Balmerterme," Z. Phys. 86, 21-23 (1933); V. Fock, "Zur Theorie des Wasserstoffatoms," ibid. 98, 145-154 (1936); M. J. Englefield, Group Theory and the Coulomb Problem (Wiley Interscience, New York, 1972); X. L. Yang, M. Lieber, and F. T. Chan, "The Runge Lenz Vector for the Two-dimensional Hydrogen Atom," Am. J. Phys. 59, 231-232 (1991).
8. M. Moshinsky, C. Quesne, and G. Loyola, "Art of Science: The determination of the symmetry Lie algebra for a Hamiltonian with accidental degeneracy," Ann. Phys. 198, 103-131 (1990).
9. See, e.g., V. I. Man'ko, "Invariants and States Generating Symmetry of Nonstationary Systems," in "Symmetries in Science, edited by B. Gruber, L. C. Biedenharn, and H. D. Doehner (Plenum, New York, 1991 ), Vol. V, pp. 453-473.
10. R. L. Liboff, Introductory Quantum Mechanics (Holden-Day), pp. 297-298, San Francisco, California, 1980.
11. See, e.g., C. Cohen-Tannoudji, Quantum Mechanics, (Wiley Interscience, New York. 1977), Vol. I.
12. W.-K. Li, "Degeneracy in the particle-in-the-square problem," Am. J. Phys. 50, 666 (1982).
13. M. Hamermesh, Group Theory and its Application to Physical Problems (Dover, New York, 1962).
14. R. W. Haymaker and R. P. Rau, "Supersymmetry in Quantum Mechanics," Am. J. Phys. 54, 928-936 (1986); F. Cooper and B. Freedman, "Aspects of Supersymmetric Quantum Mechanics," Ann. Phys. 146, 262-288 (1983); M. de Crombrugghe and V. Rittenberg, "Supersymmetric Quantum Mechanics," ibid. 151, 99-126 (1983).
15. R. W. Haymaker and R. P. Rau, "Supersymmetry in Quantum Mechanics," Am. J. Phys. 54, 928-936 (1986); F. Cooper and B. Freedman, "Aspects of Supersymmetric Quantum Mechanics," Ann. Phys. 146, 262-288 (1983); M. de Crombrugghe and V. Rittenberg, "Supersymmetric Quantum Mechanics," ibid. 151, 99-126 (1983).
16. R. W. Haymaker and R. P. Rau, "Supersymmetry in Quantum Mechanics," Am. J. Phys. 54, 928-936 (1986); F. Cooper and B. Freedman, "Aspects of Supersymmetric Quantum Mechanics," Ann. Phys. 146, 262-288 (1983); M. de Crombrugghe and V. Rittenberg, "Supersymmetric Quantum Mechanics," ibid. 151, 99-126 (1983).
17. S. L. Altmann, Induced Representations in Crystals and Molecules (Academic, New York, 1977).
18. W. Ludwig and C. Falter, Symmetries in Physics, Group Theory Applied to Physical Problems, Springer Series in Solid-State Sciences Vol. 64 (Springer, New York, 1988).
19. E. B. Bogomolny, B. Georgeot, M. J. Giannoni, and C. Schmith, "Chaotic Billiards Generated by Arithmetic Groups," Phys. Rev. Lett. 69, 1477-1480 (1992).
20. A. Frank, R. Lemus, M. Andres, and F. Leyvraz (unpublished).