On the Problem of Degeneracy in Quantum Mechanics

J. M. Jauch; E. L. Hill

doi:10.1103/PhysRev.57.641

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On the Problem of Degeneracy in Quantum Mechanics

J. M. Jauch and E. L. Hill

Department of Physics, University of Minnesota, Minneapolis, Minnesota

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Phys. Rev. 57, 641 – Published 1 April, 1940

DOI: https://doi.org/10.1103/PhysRev.57.641

Abstract

The problem of degeneracy in quantum mechanics is related to the existence of groups of contact transformations under which the Hamiltonian is invariant. The correspondence between transformations in classical and quantum theories is developed. The Fock-Bargmann treatment of the symmetry group of the hydrogenic atom comes under this theory. The symmetry group of the 2-dimensional Kepler problem is found to be the 3-dimensional rotation group; that of the $n$ -dimensional isotropic oscillator is isomorphic to the unimodular unitary group in $n$ dimensions. The 2-dimensional anisotropic oscillator has the same symmetry as the isotropic oscillator in classical mechanics, but the quantum-mechanical problem presents complications which leave its symmetry group in doubt.

References (12)

E. Wigner, Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren (Vieweg & Sohn, 1931)
V. Fock, Zeits. f. Physik 98, 145 (1935)
V. Bargmann, Zeits. f. Physik 99, 578 (1936)
H. A. Kramers, Proc. Amsterdam Acad. 33, 959 (1930) E. Wigner, Göttingen Nachrichten, Mathematisch-Physikalische Klasse (1932), p. 546
E. T. Whittaker, Analytical Dynamics (Cambridge Press, third edition), Chapters 10, 11, 12
[5], Section 144
S. Lie and F. Engel, Theorie der Transformationsgruppen (Teubner) G. Vivanti, Leçons élémentaires sur la Theorie des Groupes de Transformations (Gauthier-Villars, Paris, 1904) L. P. Eisenhart, Continuous Groups of Transformations (Princeton University Press, 1933)
P. A. M. Dirac, Quantum Mechanics (Oxford Press, second edition, 1935) E. C. Kemble, Fundamental Principles of Quantum Mechanics (McGraw-Hill, New York, 1937)
Omitted endnote
[5], Section 145
E. Hylleraas, Zeits. f. Physik 74, 216 (1932) B. Podolsky and L. Pauling, Phys. Rev. 34, 109 (1929) page 115
H. Weyl, The Classical Groups (Princeton University Press, 1939), p. 201

[c1] E. Wigner, Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren (Vieweg & Sohn, 1931)

[c2] V. Fock, Zeits. f. Physik 98, 145 (1935)

[c3] V. Bargmann, Zeits. f. Physik 99, 578 (1936)

[c4] H. A. Kramers, Proc. Amsterdam Acad. 33, 959 (1930) E. Wigner, Göttingen Nachrichten, Mathematisch-Physikalische Klasse (1932), p. 546

[c5] E. T. Whittaker, Analytical Dynamics (Cambridge Press, third edition), Chapters 10, 11, 12

[c6] [5], Section 144

[c7] S. Lie and F. Engel, Theorie der Transformationsgruppen (Teubner) G. Vivanti, Leçons élémentaires sur la Theorie des Groupes de Transformations (Gauthier-Villars, Paris, 1904) L. P. Eisenhart, Continuous Groups of Transformations (Princeton University Press, 1933)

[c8] P. A. M. Dirac, Quantum Mechanics (Oxford Press, second edition, 1935) E. C. Kemble, Fundamental Principles of Quantum Mechanics (McGraw-Hill, New York, 1937)

[c9] Omitted endnote

[c10] [5], Section 145

[c11] E. Hylleraas, Zeits. f. Physik 74, 216 (1932) B. Podolsky and L. Pauling, Phys. Rev. 34, 109 (1929) page 115

[c12] H. Weyl, The Classical Groups (Princeton University Press, 1939), p. 201

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On the Problem of Degeneracy in Quantum Mechanics

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