Exactly solvable problems of quantum mechanics and their spectrum generating algebras: A review

Central European Journal of Physics

Volumn 5, Issue 2, 2007, Pages 111-134

  Rasinariu, Constantin ^a   Mallow, Jeffry V ^b   Gangopadhyaya, Asim ^b  

^a Chair Emeritus in the Business and Entrepreneurship Department at Columbia College Chicago   (United States)

^b LOYOLA UNIVERSITY CHICAGO   (United States)

Author keywords

Potential algebra; Shape invariance; Spectrum generating algebra; Supersymmetric quantum mechanics

Indexed keywords

EID: 34147221459     PISSN: 18951082     EISSN: 16443608     Source Type: Journal    
DOI: 10.2478/s11534-007-0001-1     Document Type: Review

References (27)

1. For Dirac's ladder method, see P.A.M. Dirac: The Principles of Quantum Mechanics, 4th ed., Clarendon Press, Oxford, 1958.
2. E. Witten: "Dynamical breaking of supersymmetry", Nucl. Phys., Vol. B188, (1982), pp. 513-554; E. Witten: "Constraints on supersymmetry breaking", Nucl. Phys., Vol. B202, (1982), pp. 253-316.
3. F. Cooper and B. Freedman: "Aspects of Supersymmetric Quantum Mechanics", Ann. Phys., Vol. 146, (1983), pp. 262-288.
4. R. Dutt, A. Khare and U.P. Sukhatme: "Supersymmetry, Shape Invariance And Exactly Solvable Potentials", Am. J. Phys., Vol. 56, (1988), pp. 163-168.
5. F. Cooper, A. Khare and U. Sukhatme: Supersymmetry in Quantum Mechanics, World Scientific, New Jersey, 2001.
6. G. Junker: Supersymmetric Methods in Quantum and Statistical Physics, Springer Verlag, Berlin, 1996.
7. A.B. Balantekin: "Accidental Degeneracies and Supersymmetric Quantum Mechanics", Ann. of Phys., Vol. 164, (1985), pp. 277-287.
8. L. Infeld and T.E. Hull: "The Factorization Method", Rev. Mod. Phys., Vol. 23, (1951), pp. 21-68.
9. L.E. Gendenshtein: "Derivation of Exact Spectra of the Schrödinger Equation by Means of Supersymmetry", JETP Lett., Vol. 38, (1983), pp. 356-359; L.E.Gendenshtein and I.V. Krive: "Supersymmetry in Quantum Mechanics", Sov. Phys. Usp., Vol. 28, (1985), pp. 645-666.
10. D. Barclay, R. Dutt, A. Gangopadhyaya, A. Khare, A. Pagnamenta and U. Sukhatme: "New Exactly Solvable Hamiltonians: Shape Invariance and Self-Similarity", Phys. Rev., Vol. A48, (1993), pp. 2786-2797.
11. R. Dutt, A. Gangopadhyaya, A. Khare, A. Pagnamenta and U. Sukhatme: "Solvable quantum mechanical examples of broken supersymmetry", Phys. Lett., Vol. 174A, (1993), pp. 363-367.
12. J. Wu, Y. Alhassid and F. Gürsey: "Group theory approach to scattering. IV. Solvable potentials associated with SO(2,2)", Ann. Phys., Vol. 196, (1989), pp. 163-181.
13. J. Wu and Y. Alhassid: "The potential group approach and hypergeometric differential equations", J. Math. Phys., Vol. 31, (1990), pp. 557-562.
14. A.O. Barut, A. Inomata and R. Wilson: "A new realization of dynamical groups and factorization method", J. Phys. A: Math. Gen., Vol. 20, (1987), pp. 4075-4082; "Algebraic treatment of second Pöschl-Teller, Morse-Rosen, Eckart equations", J. Phys. A: Math. Gen., Vol. 20, (1987), pp. 4083-4096.
15. M.J. Englefield and C. Quesne: "Dynamical potential algebras for Gendenshtein and Morse potentials", J. Phys. A: Math. Gen., Vol. 24, (1991), pp. 3557-3574.
16. S.-H. Dong and Z.-Q. Ma: "The hidden symmetry for a quantum system with an infinitely deep square-well potential", Am. J. Phys., Vol. 70, (2002), pp. 520-521.
17. W. Pauli: "Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik", Z. Physik, Vol. 36, (1926), pp. 336-363; V. Fock: "Zur Theorie des Wasserstoffatoms", Z. Physik, Vol. 98, (1935), pp. 145-154
18. R. Dutt, A. Gangopadhyaya and U. Sukhatme: "Noncentral potentials and spherical harmonics using supersymmetry and shape invariance", Am. J. Phys., Vol. 65, (1996), pp. 400-403.
19. A.B. Balantekin: "Algebraic approach to shape invariance", Phys. Rev. A, Vol. 57, (1998), pp. 4188-4191.
20. A. Gangopadhyaya, J.V. Mallow and U.P. Sukhatme: "Exact solutions of the Schroedinger equation: Connection between supersymmetric quantum mechanics and spectrum generating algebras", Phys. Rev. A, Vol. 58, (1998), pp. 4287-4292.
21. A. Gangopadhyaya, J.V. Mallow, C. Rasinariu and U.P. Sukhatme: "Exact solutions of the Schroedinger equation: Connection between supersymmetric quantum mechanics and spectrum generating algebras", Chinese J. Phys., Vol. 39, (2001), pp. 101-121.
22. S. Chaturvedi, R. Dutt, A. Gangopadhyaya, P. Panigrahi, C. Rasinariu and U. Sukhatme: "Algebraic Shape Invariant Models", Phys. Lett., Vol. A248, (1998), pp. 109-113.
23. A. Gangopadhyaya, J.V. Mallow and U.P. Sukhatme: "Shape invariance and its connection to potential algebra", In: Henrik Aratyn et al. (Eds.): Proceedings of Workshop on Supersymmetric Quantum Mechanics and Integrable Models, Springer-Verlag, Berlin, 1997.
24. R. Dutt, A. Gangopadhyaya, C. Rasinariu and U. Sukhatme: "Coordinate Realizations of Deformed Lie Algebras with Three Generators", Phys. Rev., Vol. A60, (1999), pp. 3482-3486.
25. For a very readable introduction to so(2, 1)-algebra and its representation, see B.G. Adams, J. Cizeka and J. Paldus: "Lie Algebraic Methods And Their Applications to Simple Quantum Systems", In: P-O. Löwdin (Ed.): Advances in Quantum Chemistry, Vol. 19, Academic Press, New York, 1987.
26. M. Rocek: "Representation theory of the nonlinear SU (2) algebra", Phys. Lett. B, Vol. 255, (1991), pp. 554-557.
27. A. Gangopadhyaya, J.V. Mallow and U.P. Sukhatme: "Broken Supersymmetric Shape Invariant Systems and Their Potential Algebras", Phys. Lett., Vol. A283, (2001), pp. 279-284.