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


SIGMA 5 (2009), 046, 17 pages      arXiv:0807.4650      https://doi.org/10.3842/SIGMA.2009.046

Point Canonical Transformation versus Deformed Shape Invariance for Position-Dependent Mass Schrödinger Equations

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 February 09, 2009, in final form March 19, 2009; Published online April 15, 2009

Abstract
On using the known equivalence between the presence of a position-dependent mass (PDM) in the Schrödinger equation and a deformation of the canonical commutation relations, a method based on deformed shape invariance has recently been devised for generating pairs of potential and PDM for which the Schrödinger equation is exactly solvable. This approach has provided the bound-state energy spectrum, as well as the ground-state and the first few excited-state wavefunctions. The general wavefunctions have however remained unknown in explicit form because for their determination one would need the solutions of a rather tricky differential-difference equation. Here we show that solving this equation may be avoided by combining the deformed shape invariance technique with the point canonical transformation method in a novel way. It consists in employing our previous knowledge of the PDM problem energy spectrum to construct a constant-mass Schrödinger equation with similar characteristics and in deducing the PDM wavefunctions from the known constant-mass ones. Finally, the equivalence of the wavefunctions coming from both approaches is checked.

Key words: Schrödinger equation; position-dependent mass; shape invariance; point canonical transformations.

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References

  1. Ring P., Schuck P., The nuclear many-body problem, Texts and Monographs in Physics, Springer-Verlag, New York - Berlin, 1980.
  2. Luttinger J.M., Kohn W., Motion of electrons and holes in perturbed periodic fields, Phys. Rev. 97 (1955), 869-883.
  3. Harrison P., Quantum wells, wires and dots, Wiley, Chichester, 1999.
  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. Weisbuch C., Vinter B., Quantum semiconductor heterostructures, Academic, New York, 1997.
  6. Chamel N., Effective mass of free neutrons in neutron star crust, Nuclear Phys. A 773 (2006), 263-278, nucl-th/0512034.
  7. Quesne C., Tkachuk V.M., Deformed algebras, position-dependent effective masses and curved spaces: an exactly solvable Coulomb problem, J. Phys. A: Math. Gen. 37 (2004), 4267-4281, math-ph/0403047.
  8. Infeld I., Schild A., A note on the Kepler problem in a space of constant negative curvature, Phys. Rev. 67 (1945), 121-122.
  9. Cariñena J.F., Rañada M.F., Santander M., A quantum exactly solvable non-linear oscillator with quasi-harmonic behaviour, Ann. Physics 322 (2007), 434-459, math-ph/0604008.
  10. Bagchi B., Quesne C., Roychoudhury R., Pseudo-Hermitian versus Hermitian position-dependent-mass Hamiltonians in a perturbative framework, J. Phys. A: Math. Gen. 39 (2006), L127-L134, quant-ph/0511182.
  11. Bagchi B., Banerjee A., Quesne C., PT-symmetric quartic anharmonic oscillator and position-dependent mass in a perturbative approach, Czech. J. Phys. 56 (2006), 893-898, quant-ph/0606012.
  12. Bender C.M., Making sense of non-Hermitian Hamiltonians, Rep. Progr. Phys. 70 (2007), 947-1018, hep-th/0703096.
  13. von Roos O., Position-dependent effective masses in semiconductor theory, Phys. Rev. B 27 (1983), 7547-7552.
  14. Gritsev V.V., Kurochkin Yu.A., Model of excitations in quantum dots based on quantum mechanics in spaces of constant curvature, Phys. Rev. B 64 (2001), 035308, 9 pages.
  15. Bagchi B., Banerjee A., Quesne C., Tkachuk V.M., Deformed shape invariance and exactly solvable Hamiltonians with position-dependent effective mass, J. Phys. A: Math. Gen. 38 (2005), 2929-2945, quant-ph/0412016.
  16. Quesne C., Oscillator-Morse-Coulomb mappings and algebras for constant or position-dependent mass, J. Math. Phys. 49 (2008), 022106, 15 pages, arXiv:0712.1965.
  17. Cooper F., Khare A., Sukhatme U., Supersymmetry and quantum mechanics, Phys. Rep. 251 (1995), 267-385, hep-th/9405029.
  18. Junker G., Supersymmetric methods in quantum and statistical physics, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1996.
  19. Bagchi B., Supersymmetry in quantum and classical physics, Chapman and Hall/CRC, Boca Raton, FL, 2000.
  20. Spiridonov V., Exactly solvable potentials and quantum algebras, Phys. Rev. Lett. 69 (1992), 398-401, hep-th/9112075.
  21. Spiridonov V., Deformed conformal and supersymmetric quantum mechanics, Modern Phys. Lett. A 7 (1992), 1241-1251, hep-th/9202013.
  22. Khare A., Sukhatme U.P., New shape-invariant potentials in supersymmetric quantum mechanics, J. Phys. A: Math. Gen. 26 (1993), L901-L904, hep-th/9212147.
  23. Barclay D.T., Dutt R., Gangopadhyaya A., Khare A., Pagnamenta A., Sukhatme U., New exactly solvable Hamiltonians: shape invariance and self-similarity, Phys. Rev. A 48 (1993), 2786-2797, hep-ph/9304313.
  24. Sukhatme U.P., Rasinariu C., Khare A., Cyclic shape invariant potentials, Phys. Lett. A 234 (1997), 401-409, hep-ph/9706282.
  25. Gangopadhyaya A., Mallow J.V., Rasinariu C., Sukhatme U.P., Exactly solvable models in supersymmetric quantum mechanics and connection with spectrum-generating algebras, Theoret. and Math. Phys. 118 (1999), 285-294, hep-th/9810074.
  26. Loutsenko I., Spiridonov V., Vinet L., Zhedanov A., Spectral analysis of q-oscillator with general bilinear interaction, J. Phys. A: Math. Gen. 31 (1998), 9081-9094.
  27. Quesne C., Spectrum generating algebras for position-dependent mass oscillator Schrödinger equations, J. Phys. A: Math. Theor. 40 (2007), 13107-13120, arXiv:0705.0862.
  28. Bhattacharjie A., Sudarshan E.C.G., A class of solvable potentials, Nuovo Cimento 25 (1962), 864-879.
  29. Lévai G., A search for shape-invariant solvable potentials, J. Phys. A: Math. Gen. 22 (1989), 689-702.
  30. De R., Dutt R., Sukhatme U., Mapping of shape invariant potentials under point canonical transformations, J. Phys. A: Math. Gen. 25 (1992), L843-L850.
  31. 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.
  32. Cordero P., Salamó S., Algebraic solution for the Natanzon confluent potentials, J. Phys. A: Math. Gen. 24 (1991), 5299-5306.
  33. BenDaniel D.J., Duke C.B., Space-charge effects on electron tunneling, Phys. Rev. 152 (1966), 683-692.
  34. Lévy-Leblond J.-M., Position-dependent effective mass and Galilean invariance, Phys. Rev. A 52 (1995), 1845-1849.
  35. Bagchi B., Gorain P., Quesne C., Roychoudhury R., A general scheme for the effective-mass Schrödinger equation and the generation of the associated potentials, Modern Phys. Lett. A 19 (2004), 2765-2775, quant-ph/0405193.
  36. Alhaidari A.D., Solutions of the nonrelativistic wave equation with position-dependent effective mass, Phys. Rev. A 66 (2002), 042116, 7 pages, quant-ph/0207061.
  37. Bagchi B., Gorain P., Quesne C., Roychoudhury R., New approach to (quasi-)exactly solvable Schrödinger equations with a position-dependent effective mass, Europhys. Lett. 72 (2005), 155-161, quant-ph/0505171.
  38. Quesne C., Application of nonlinear deformation algebra to a physical system with Pöschl-Teller potential, J. Phys. A: Math. Gen. 32 (1999), 6705-6710, math-ph/9911004.
  39. Koekoek R., Swarttouw R.F., The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, math.CA/9602214.
  40. Abramowitz M., Stegun I.A., Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, Vol. 55, Washington, D.C., 1964.
  41. Romanovski V., Sur quelques classes nouvelles de polynômes orthogonaux, C. R. Acad. Sci. Paris 188 (1929), 1023-1025.
  42. Compean C.B., Kirchbach M., The trigonometric Rosen-Morse potential in the supersymmetric quantum mechanics and its exact solutions, J. Phys. A: Math. Gen. 39 (2006), 547-558, quant-ph/0509055.
  43. Raposo A.P., Weber H.J., Alvarez-Castillo D., Kirchbach M., Romanovski polynomials in selected physics problems, Cent. Eur. J. Phys. 5 (2007), 253-284, arXiv:0706.3897.
  44. Gradshteyn I.S., Ryzhik I.M., Table of integrals, series, and products, Academic Press, New York, 1980.

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