Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 3 (2007), 067, 14 pages      arXiv:0705.2577      https://doi.org/10.3842/SIGMA.2007.067

Quadratic Algebra Approach to an Exactly Solvable Position-Dependent Mass Schrödinger Equation in Two Dimensions

Christiane Quesne
Physique Nucléaire Théorique et Physique Mathématique, Université Libre de Bruxelles, Campus de la Plaine CP229, Boulevard du Triomphe, B-1050 Brussels, Belgium

Received March 30, 2007, in final form May 08, 2007; Published online May 17, 2007

Abstract
An exactly solvable position-dependent mass Schrödinger equation in two dimensions, depicting a particle moving in a semi-infinite layer, is re-examined in the light of recent theories describing superintegrable two-dimensional systems with integrals of motion that are quadratic functions of the momenta. To get the energy spectrum a quadratic algebra approach is used together with a realization in terms of deformed parafermionic oscillator operators. In this process, the importance of supplementing algebraic considerations with a proper treatment of boundary conditions for selecting physical wavefunctions is stressed. Some new results for matrix elements are derived. This example emphasizes the interest of a quadratic algebra approach to position-dependent mass Schrödinger equations.

Key words: Schrödinger equation; position-dependent mass; quadratic algebra.

pdf (251 kb)   ps (164 kb)   tex (19 kb)

References

  1. Bastard G., Wave mechanics applied to semiconductor heterostructures, Editions de Physique, Les Ulis, 1988.
  2. Serra L., Lipparini E., Spin response of unpolarized quantum dots, Europhys. Lett. 40 (1997), 667-672.
  3. Ring P., Schuck P., The nuclear many body problem, Springer, New York, 1980.
  4. Arias de Saavedra F., Boronat J., Polls A., Fabrocini A., Effective mass of one 4He atom in liquid 3He, Phys. Rev. B 50 (1994), 4248-4251, cond-mat/9403075.
  5. Barranco M., Pi M., Gatica S.M., Hernández E.S., Navarro J., Structure and energetics of mixed 4He-3He drops, Phys. Rev. B 56 (1997), 8997-9003.
  6. Puente A., Serra Ll., Casas M., Dipole excitation of Na clusters with a non-local energy density functional, Z. Phys. D 31 (1994), 283-286.
  7. Quesne C., First-order intertwining operators and position-dependent mass Schrödinger equations in d dimensions, Ann. Physics 321 (2006), 1221-1239, quant-ph/0508216.
  8. Bhattacharjie A., Sudarshan E.C.G., A class of solvable potentials, Nuovo Cimento 25 (1962), 864-879.
  9. Natanzon G.A., General properties of potentials for which the Schrödinger equation can be solved by means of hypergeometric functions, Theoret. and Math. Phys. 38 (1979), 146-153.
  10. Lévai G., A search for shape-invariant solvable potentials, J. Phys. A: Math. Gen. 22 (1989), 689-702.
  11. Alhassid Y., Gürsey F., Iachello F., Group theory approach to scattering. II. The Euclidean connection, Ann. Physics 167 (1986), 181-200.
  12. Wu J., Alhassid Y., The potential group approach and hypergeometric differential equations, J. Math. Phys. 31 (1990), 557-562.
  13. Englefield M.J., Quesne C., Dynamical potential algebras for Gendenshtein and Morse potentials, J. Phys. A: Math. Gen. 24 (1991), 3557-3574.
  14. Lévai G., Solvable potentials associated with su(1,1) algebras: a systematic study, J. Phys. A: Math. Gen. 27 (1994), 3809-3828.
  15. Cooper F., Khare A., Sukhatme U., Supersymmetry and quantum mechanics, Phys. Rep. 251 (1995), 267-385, hep-th/9405029.
  16. Bagchi B., Supersymmetry in quantum and classical mechanics, Chapman and Hall/CRC, Boca Raton, FL, 2000.
  17. Chen G., Chen Z., Exact solutions of the position-dependent mass Schrödinger equation in D dimensions, Phys. Lett. A 331 (2004), 312-315.
  18. Dong S.-H., Lozada-Cassou M., Exact solutions of the Schrödinger equation with the position-dependent mass for a hard-core potential, Phys. Lett. A 337 (2005), 313-320.
  19. Mustafa O., Mazharimousavi S.H., d-dimensional generalization of the point canonical transformation for a quantum particle with position-dependent mass, J. Phys. A: Math. Gen. 39 (2006), 10537-10547, math-ph/0602044.
  20. Mustafa O., Mazharimousavi S.H., Quantum particles trapped in a position-dependent mass barrier; a d-dimensional recipe, Phys. Lett. A 358 (2006), 259-261, quant-ph/0603134.
  21. Ju G.-X., Xiang Y., Ren Z.-Z., The localization of s-wave and quantum effective potential of a quasi-free particle with position-dependent mass, quant-ph/0601005.
  22. Gönül B., Koçak M., Explicit solutions for N-dimensional Schrödinger equations with position-dependent mass, J. Math. Phys. 47 (2006), 102101, 6 pages, quant-ph/0512035.
  23. Olendski O., Mikhailovska L., Bound-state evolution in curved waveguides and quantum wires, Phys. Rev. B 66 (2002), 035331, 8 pages.
  24. Gudmundsson V., Tang C.-S., Manolescu A., Bound state with negative binding energy induced by coherent transport in a two-dimensional quantum wire, Phys. Rev. B 72 (2005), 153306, 4 pages, cond-mat/0506009.
  25. Goldstein H., Classical mechanics, Addison-Wesley, Reading, MA, 1980.
  26. Dirac P.A.M., The principles of quantum mechanics, Oxford University Press, Oxford, 1981.
  27. Fris I., Mandrosov V., Smorodinsky Ya.A., Uhlir M., Winternitz P., On higher symmetries in quantum mechanics, Phys. Lett. 16 (1965), 354-356.
  28. Winternitz P., Smorodinsky Ya.A., Uhlir M., Fris I., Symmetry groups in classical and quantum mechanics, Sov. J. Nucl. Phys. 4 (1967), 444-450.
  29. Makharov A.A., Smorodinsky Ya.A., Valiev Kh., Winternitz P., A systematic search for nonrelativistic systems with dynamical symmetries. Part I: the integrals of motion, Nuovo Cimento A 52 (1967), 1061-1084.
  30. Hietarinta J., Direct methods for the search of the second invariant, Phys. Rep. 147 (1987), 87-154.
  31. Granovskii Ya.I., Lutzenko I.M., Zhedanov A.S., Mutual integrability, quadratic algebras, and dynamical symmetry, Ann. Physics 217 (1992), 1-20.
  32. Zhedanov A.S., "Hidden symmetry" of Askey-Wilson polynomials, Theoret. and Math. Phys. 89 (1991), 1146-1157.
  33. Granovskii Ya.I., Zhedanov A.S., Lutsenko I.M., Quadratic algebras and dynamics in curved spaces. I. Oscillator, Theoret. and Math. Phys. 91 (1992), 474-480.
  34. Granovskii Ya.I., Zhedanov A.S., Lutsenko I.M., Quadratic algebras and dynamics in curved spaces. II. The Kepler problem, Theoret. and Math. Phys. 91 (1992), 604-612.
  35. Bonatsos D., Daskaloyannis C., Kokkotas K., Deformed oscillator algebras for two-dimensional quantum superintegrable systems, Phys. Rev. A 50 (1994), 3700-3709, hep-th/9309088.
  36. Daskaloyannis C., Quadratic Poisson algebras of two-dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems, J. Math. Phys. 42 (2001), 1100-1119, math-ph/0003017.
  37. Daskaloyannis C., Ypsilantis K., Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two-dimensional manifold, J. Math. Phys. 47 (2006), 042904, 38 pages, math-ph/0412055.
  38. Daskaloyannis C., Tanoudes Y., Classification of quantum superintegrable systems with quadratic integrals on two dimensional manifolds, math-ph/0607058.
  39. Létourneau P., Vinet L., Superintegrable systems: polynomial algebras and quasi-exactly solvable Hamiltonians, Ann. Physics 243 (1995), 144-168.
  40. Rañada M.F., Superintegrable n = 2 systems, quadratic constants of motion, and potentials of Drach, J. Math. Phys. 38 (1997), 4165-4178.
  41. Rañada M.F., Santander M., Superintegrable systems on the two-dimensional sphere S2 and the hyperbolic plane H2, J. Math. Phys. 40 (1999), 5026-5057.
  42. Tempesta P., Turbiner A.V., Winternitz P., Exact solvability of superintegrable systems, J. Math. Phys. 42 (2001), 4248-4257, hep-th/0011209.
  43. Kalnins E.G., Miller W.Jr., Pogosyan G.S., Superintegrability on the two-dimensional hyperboloid, J. Math. Phys. 38 (1997), 5416-5433.
  44. Kalnins E.G., Miller W.Jr., Hakobyan Y.M., Pogosyan G.S., Superintegrability on the two-dimensional hyperboloid. II, J. Math. Phys. 40 (1999), 2291-2306, quant-ph/9907037.
  45. Kalnins E.G., Kress J.M., Miller W.Jr., Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory, J. Math. Phys. 46 (2005), 053509, 28 pages.
  46. Kalnins E.G., Kress J.M., Miller W.Jr., Second-order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform, J. Math. Phys. 46 (2005), 053510, 15 pages.
  47. Kalnins E.G., Kress J.M., Miller W.Jr., Second-order superintegrable systems in conformally flat spaces. V. Two- and three-dimensional quantum systems, J. Math. Phys. 47 (2006), 093501, 25 pages.
  48. Kalnins E.G., Kress J.M., Miller W.Jr., Nondegenerate 2D complex Euclidean superintegrable systems and algebraic varieties, J. Phys. A: Math. Theor. 40 (2007), 3399-3411.
  49. Quesne C., Generalized deformed parafermions, nonlinear deformations of so(3) and exactly solvable potentials, Phys. Lett. A 193 (1994), 245-250.
  50. Daskaloyannis C., Generalized deformed oscillator and nonlinear algebras, J. Phys. A: Math. Gen. 24 (1991), L789-L794.
  51. Roy B., Roy P., Effective mass Schrödinger equation and nonlinear algebras, Phys. Lett. A 340 (2005), 70-73.
  52. von Roos O., Position-dependent effective masses in semiconductor theory, Phys. Rev. B 27 (1983), 7547-7552.

Previous article   Next article   Contents of Volume 3 (2007)