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

SIGMA 6 (2010), 088, 8 pages      arXiv:1003.3003     doi:10.3842/SIGMA.2010.088

Flatland Position-Dependent-Mass: Polar Coordinates, Separability and Exact Solvability

S. Habib Mazharimousavi and Omar Mustafa
Department of Physics, Eastern Mediterranean University, G Magusa, North Cyprus, Mersin 10, Turkey

Received August 15, 2010, in final form October 26, 2010; Published online October 29, 2010

Abstract
The kinetic energy operator with position-dependent-mass in plane polar coordinates is obtained. The separability of the corresponding Schrödinger equation is discussed. A hypothetical toy model is reported and two exactly solvable examples are studied.

Key words: position dependent mass; polar coordinates; separability; exact solvability.

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References

  1. Puente A., Casas M., Non-local energy density functional for atoms and metal clusters, Comput. Mater Sci. 2 (1994), 441-449.
  2. Plastino A.R., Casas M., Plastino A., Bohmian quantum theory of motion for particles with position-dependent effective mass, Phys. Lett. A 281 (2001), 297-304.
  3. Schmidt A.G.M., Wave-packet revival for the Schrödinger equation with position-dependent mass, Phys. Lett. A 353 (2006), 459-462.
  4. 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.
  5. Vakarchuk I.O., The Kepler problem in Dirac theory for a particle with position-dependent mass, J. Phys. A: Math. Gen. 38 (2005), 4727-4734, quant-ph/0502105.
  6. Cai C.-Y., Ren Z.-Z., Ju G.-X., Exact solutions to three-dimensional Schrödinger equation with an exponentially position-dependent mass, Commun. Theor. Phys. (Beijing) 43 (2005), 1019-1022.
  7. Roy B., Roy P., Effective mass Schrödinger equation and nonlinear algebras, Phys. Lett. A 340 (2005), 70-73.
  8. Gönül B., Koçak M., Remarks on exact solvability of quantum systems with spatially varying effective mass, Chinese Phys. Lett. 20 (2005), 2742-2745.
  9. de Souza Dutra A., Almeida C.A.S., Exact solvability of potentials with spatially dependent effective masses, Phys Lett. A 275 (2000), 25-30.
  10. Mustafa O., Mazharimousavi S.H., Ordering ambiguity revisited via position dependent mass pseudo-momentum operators, Internat. J. Theoret. Phys. 46 (2007), 1786-1796, quant-ph/0607158.
  11. Cruz y Cruz S., Negro J., Nieto L.M., Classical and quantum position-dependent mass harmonic oscillators, Phys. Lett. A 369 (2007), 400-406.
  12. Cruz y Cruz S., Rosas-Ortiz O., Position-dependent mass oscillators and coherent states, J. Phys. A: Math. Theor. 42 (2009), 185205, 21 pages.
  13. Lekner J., Reflectionless eigenstates of the sech2 potential, Amer. J. Phys. 75 (2007), 1151-1157.
  14. 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.
  15. Jiang L., Yi L.-Z., Jia C.-S., Exact solutions of the Schrödinger equation with position-dependent mass for some Hermitian and non-Hermitian potentials, Phys. Lett. A 345 (2005), 279-286.
  16. 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.
  17. Diaz J.I., Negro J., Nieto L.M., Rosas-Ortiz O., The supersymmetric modified Pöschl-Teller and delta well potentials, J. Phys. A: Math. Gen. 32 (1999), 8447-8460, quant-ph/9910017.
  18. 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.
    Gritsev V.V., Kurochkin Y.A., Model of excitations in quantum dots based on quantum mechanics in spaces of constant curvature, Phys. Rev. B 64 (2001), 035308, 9 pages.
  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.
    Lévai G., Özer O., An exactly solvable Schrödinger equation with finite positive position-dependent effective mass, J. Math. Phys. 51 (2010), 092103, 13 pages.
  20. 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.
    Bagchi B., Ganguly A., Sinha A., Supersymmetry across nanoscale heterojunction, Phys. Lett. A 374 (2010), 2397-2400, arXiv:1002.2732.
  21. Yu J., Dong S.-H., Exactly solvable potentials for the Schrödinger equation with spatially dependent mass, Phys. Lett. A 325 (2004), 194-198.
  22. Quesne C., First-order intertwining operators and position-dependent mass Schrödinger equations in d dimensions, Ann. Physics 321 (2006), 1221-1239, quant-ph/0508216.
  23. Tanaka T., N-fold supersymmetry in quantum systems with position-dependent mass, J. Phys. A: Math. Gen. 39 (2006), 219-234, quant-ph/0509132.
  24. de Souza Dutra A., Ordering ambiguity versus representation, J. Phys. A: Math. Gen. 39 (2006), 203-208, arXiv:0705.3247.
  25. von Roos O., Position-dependent effective masses in semiconductor theory, Phys. Rev. B 27 (1983), 7547-7552.
    Lévy-Leblond J.M., Position-dependent effective mass and Galilean invariance, Phys. Rev. A 52 (1995), 1845-1849.
  26. Mustafa O., Mazharimousavi S.H., Non-Hermitian d-dimensional Hamiltonians with position-dependent mass and their η-pseudo-Hermiticity generators, Czechoslovak J. Phys. 56 (2006), 967-975, quant-ph/0603272.
  27. Mustafa O., Mazharimousavi S.H., η-weak-pseudo-Hermiticity generators and exact solvability, Phys. Lett. A 357 (2006), 295-297, quant-ph/0604106.
  28. Mustafa O., Mazharimousavi S.H., Complexified von Roos Hamiltonian's η-weak-pseudo-Hermiticity, isospectrality and exact solvability, J. Phys. A: Math. Theor. 41 (2008), 244020, 8 pages, arXiv:0707.3738.
  29. Mustafa O., Mazharimousavi S.H., A singular position-dependent mass particle in an infinite potential well, Phys. Lett. A 373 (2009), 325-327, arXiv:0807.3030.
  30. Mustafa O., The shifted-1/N-expansion method for two-dimensional hydrogenic donor states in an arbitrary magnetic field, J. Phys.: Condens. Matter 5 (1993), 1327-1332.

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