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

• Authors: Constantin Rasinariu , Jeffry Mallow , Asim Gangopadhyaya
• Languages of publication: EN

Abstracts

EN

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.

Keywords

EN

Supersymmetric quantum mechanics; shape invariance; potential algebra; spectrum-generating algebra

Discipline

• 03.65.-w: Quantum mechanics[see also 03.67.-a Quantum information; 05.30.-d Quantum statistical mechanics; 31.30.J- Relativistic and quantum electrodynamics (QED) effects in atoms, molecules, and ions in atomic physics]
• 11.30.Pb: Supersymmetry(see also 12.60.Jv Supersymmetric models)
• 03.65.Fd: Algebraic methods(see also 02.20.-a Group theory)

• Publisher: De Gruyter Open
• Journal: Open Physics
• Year: 2007
• Volume: 5
• Issue: 2
• Pages: 111-134

Dates

published

1 - 6 - 2007

online

2 - 2 - 2007

Contributors

author

Constantin Rasinariu

• Department of Science and Mathematics, Columbia College Chicago, Chicago, USA

author

Jeffry Mallow

• Department of Physics, Loyola University Chicago, Chicago, USA

author

Asim Gangopadhyaya

• Department of Physics, Loyola University Chicago, Chicago, USA

References

• [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]
• [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. http://dx.doi.org/10.1016/0003-4916(85)90017-X[Crossref]
• [8] L. Infeld and T.E. Hull: “The Factorization Method”, Rev. Mod. Phys., Vol. 23, (1951), pp. 21–68. http://dx.doi.org/10.1103/RevModPhys.23.21[Crossref]
• [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. 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.

Identifiers

DOI

10.2478/s11534-007-0001-1