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SIGMA 6 (2010), 088, 8 pages arXiv:1003.3003
https://doi.org/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
- Puente A., Casas M.,
Non-local energy density functional for atoms and metal clusters,
Comput. Mater Sci. 2 (1994), 441-449.
- 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.
- Schmidt A.G.M.,
Wave-packet revival for the Schrödinger equation with position-dependent mass,
Phys. Lett. A 353 (2006), 459-462.
- 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.
- 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.
- 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.
- Roy B., Roy P.,
Effective mass Schrödinger equation and nonlinear algebras,
Phys. Lett. A 340 (2005), 70-73.
- Gönül B., Koçak M.,
Remarks on exact solvability of quantum systems with spatially varying effective mass,
Chinese Phys. Lett. 20 (2005), 2742-2745.
- de Souza Dutra A., Almeida C.A.S.,
Exact solvability of potentials with spatially dependent effective masses,
Phys Lett. A 275 (2000), 25-30.
- 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.
- Cruz y Cruz S., Negro J., Nieto L.M.,
Classical and quantum position-dependent mass harmonic oscillators,
Phys. Lett. A 369 (2007), 400-406.
- Cruz y Cruz S., Rosas-Ortiz O.,
Position-dependent mass oscillators and coherent states,
J. Phys. A: Math. Theor. 42 (2009), 185205, 21 pages.
- Lekner J.,
Reflectionless eigenstates of the sech2 potential,
Amer. J. Phys. 75 (2007), 1151-1157.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- Yu J., Dong S.-H.,
Exactly solvable potentials for the Schrödinger equation with spatially dependent mass,
Phys. Lett. A 325 (2004), 194-198.
- Quesne C.,
First-order intertwining operators and position-dependent mass Schrödinger equations in d dimensions,
Ann. Physics 321 (2006), 1221-1239,
quant-ph/0508216.
- Tanaka T.,
N-fold supersymmetry in quantum systems with position-dependent mass,
J. Phys. A: Math. Gen. 39 (2006), 219-234,
quant-ph/0509132.
- de Souza Dutra A.,
Ordering ambiguity versus representation,
J. Phys. A: Math. Gen. 39 (2006), 203-208,
arXiv:0705.3247.
- 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.
- 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.
- Mustafa O., Mazharimousavi S.H.,
η-weak-pseudo-Hermiticity generators and exact solvability,
Phys. Lett. A 357 (2006), 295-297,
quant-ph/0604106.
- 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.
- 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.
- 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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