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2007 | 5 | 2 | 111-134

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


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In this review, we summarize the progress that has been made in connecting supersymmetry and spectrum generating algebras through the property of shape invariance. This monograph is designed to be used by our fellow researchers, by other interested physicists, and by students at the graduate and even undergraduate levels who would like a brief introduction to the field.


  • Department of Science and Mathematics, Columbia College Chicago, Chicago, USA
  • Department of Physics, Loyola University Chicago, Chicago, USA
  • Department of Physics, Loyola University Chicago, Chicago, USA


  • [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. http://dx.doi.org/10.1016/0003-4916(83)90034-9[Crossref]
  • [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. http://dx.doi.org/10.1119/1.15697[Crossref]
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  • [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.
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  • [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. http://dx.doi.org/10.1016/0003-4916(89)90049-3[Crossref]
  • [13] J. Wu and Y. Alhassid: “The potential group approach and hypergeometric differential equations”, J. Math. Phys., Vol. 31, (1990), pp. 557–562. http://dx.doi.org/10.1063/1.528889[Crossref]
  • [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. http://dx.doi.org/10.1088/0305-4470/20/13/016[Crossref]
  • [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. http://dx.doi.org/10.1088/0305-4470/24/15/023[Crossref]
  • [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. http://dx.doi.org/10.1119/1.1456073[Crossref]
  • [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 http://dx.doi.org/10.1007/BF01450175[Crossref]
  • [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. http://dx.doi.org/10.1119/1.18551[Crossref]
  • [19] A.B. Balantekin: “Algebraic approach to shape invariance”, Phys. Rev. A, Vol. 57, (1998), pp. 4188–4191. http://dx.doi.org/10.1103/PhysRevA.57.4188[Crossref]
  • [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. http://dx.doi.org/10.1103/PhysRevA.58.4287[Crossref]
  • [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. http://dx.doi.org/10.1016/0370-2693(91)90265-R[Crossref]
  • [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.

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