A New Class of Exactly Solvable Models within the Schrödinger Equation with Position Dependent Mass

Anis Dhahbi¹, Yassine Chargui¹, Adel Trablesi²

¹Physics Department, College of Science and Arts at ArRass, Qassim University, Al-Mulida, KSA.
²Université de Tunis El Manar, Faculté des Sciences de Tunis, Unité de Recherche de Physique Nucléaire et des Hautes Energies, Tunis, Tunisie.

DOI: 10.4236/jamp.2019.75068                

Abstract

The study of physical systems endowed with a position-dependent mass (PDM) remains a fundamental issue of quantum mechanics. In this paper we use a new approach, recently developed by us for building the quantum kinetic energy operator (KEO) within the Schrodinger equation, in order to construct a new class of exactly solvable models with a position varying mass, presenting a harmonic-oscillator-like spectrum. To do so we utilize the formalism of supersymmetric quantum mechanics (SUSY QM) along with the shape invariance condition. Recent outcomes of non-Hermitian quantum mechanics are also taken into account.

Keywords

Schrödinger Equation, Position Dependent Mass, Kinetic Energy Operator, Solvable Models, Supersymmetric Quantum Mechanics, Shape Invariance

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References

[1]	Geller, M. and Kohn, W. (1993) Quantum Mechanics of Electrons in Crystals with Graded Composition. Physical Review Letters, 70, 3103. https://doi.org/10.1103/PhysRevLett.70.3103
[2]	Ring, P. and Schuck, P. (1980) The Nuclear Many-Body Problem. Springer, Berlin.
[3]	Ganguly, A., Kuru, S., Negro, J. and Nieto, L. (2006) A Study of the Bound States for Square Potential Wells with Position-Dependent Mass. Physics Letters A, 360, 228-233. https://doi.org/10.1016/j.physleta.2006.08.032
[4]	Plastino, A., Casas, M. and Plastino, A. (2001) Bohmian Quantum Theory of Motion for Particles with Position-Dependent Effective Mass. Physics Letters A, 281, 297-304. https://doi.org/10.1016/S0375-9601(01)00143-8
[5]	de Saavedra, F., Boronat, J., Polls, A. and Fabrocini, A. (1994) E ective Mass of One 4He Atom in Liquid 3He. Physical Review B, 50, 4248(R). https://doi.org/10.1103/PhysRevB.50.4248
[6]	Puente, A., Serra, L. and Casas, M. (1994) Dipole Excitation of Na Clusters with a Non-Local Energy Density Functional. Zeitschrift fur Physik D Atoms, Molecules and Clusters, 31, 283-286. https://doi.org/10.1007/BF01445008
[7]	BenDaniel, D. and Duke, C. (1966) Space-Charge E ects on Electron Tunneling. Physical Review, 152, 683. https://doi.org/10.1103/PhysRev.152.683
[8]	Li, M.T. and Kuhn, K. (1993) Band-O set Ratio Dependence on the E ective-Mass Hamiltonian Based on a Modi ed Pro le of the GaAs-Alx-Ga1-xAs Quantum Well. Physical Review B, 47, Article ID: 12760. https://doi.org/10.1103/PhysRevB.47.12760
[9]	Bastard, G. (1981) Superlattice Band Structure in the Envelope-Function Approximation. Physical Review B, 24, 5693. https://doi.org/10.1103/PhysRevB.24.5693
[10]	Gora, T. and Williams, F. (1969) Theory of Electronic States and Transport in Graded Mixed Semiconductors. Physical Review, 177, 1179. https://doi.org/10.1103/PhysRev.177.1179
[11]	Zhu, Q. and Kroemer, H. (1983) Interface Connection Rules for E ective-Mass Wave Functions at an Abrupt Heterojunction between Two Di erent Semiconductors. Physical Review B, 27, 3519. https://doi.org/10.1103/PhysRevB.27.3519
[12]	Morrow, R. and Brownstein, K. (1984) Model E ective-Mass Hamiltonians for Abrupt Heterojunctions and the Associated Wave-Function-Matching Conditions. Physical Review B, 30, 678. https://doi.org/10.1103/PhysRevB.30.678
[13]	Bastard, G. (1992) Wave Mechanics Applied to Semiconductor Hetero Structures. EDP Sciences. Les Editions de Physique, Les Ulis, France.
[14]	Von Roos, O. (1983) Position-Dependent E ective Masses in Semiconductor Theory. Physical Review B, 27, 7547. https://doi.org/10.1103/PhysRevB.27.7547
[15]	Von Roos, O. (1985) Position-Dependent E ective Masses in Semiconductor Theory. II. Physical Review B, 31, 2294. https://doi.org/10.1103/PhysRevB.31.2294
[16]	Morrow, R. (1987) Establishment of an E ective-Mass Hamiltonian for Abrupt Heterojunctions. Physical Review B, 35, 8074. https://doi.org/10.1103/PhysRevB.35.8074
[17]	Harrison, P. (2000) Quantum Wells, Wires and Dots: Theoretical and Computational Physics of Semiconductor Nanostructures. Wiley, New York.
[18]	Boztosun, I., Bonatsos, D. and Inci, I. (2008) Analytical Solutions of the Bohr Hamiltonian with the Morse Potential. Physical Review C, 77, Article ID: 044302. https://doi.org/10.1103/PhysRevC.77.044302
[19]	Bonatsos, D., Georgoudis, P., Lenis, D., Minkov, N. and Quesne, C. (2011) Bohr Hamiltonian with a Deformation-Dependent Mass Term for the Davidson Potential. Physical Review C, 83, Article ID: 044321. https://doi.org/10.1103/PhysRevC.83.044321
[20]	Hamdouni, Y. (2011) Motion of Position-Dependent E ective Mass as a Damping-Antidamping Process: Application to the Fermi Gas and to the Morse Potential. Journal of Physics A: Mathematical and Theoretical, 44, Article ID: 385301. https://doi.org/10.1088/1751-8113/44/38/385301
[21]	Mustafa, O. (2011) Radial Power-Law Position-Dependent Mass: Cylindrical Coordinates, Separability and Spectral Signatures. Journal of Physics A: Mathematical and Theoretical, 44, Article ID: 355303. https://doi.org/10.1088/1751-8113/44/35/355303
[22]	Foulkes, W. and Schluter, M. (1990) Pseudopotentials with Position-Dependent Electron Masses. Physical Review B, 42, Article ID: 11505. https://doi.org/10.1103/PhysRevB.42.11505
[23]	Barranco, M., Pi, M., Gatica, S., Hernandez, E. and Navarro, J. (1997) Structure and Energetics of Mixed 4He-3He Drops. Physical Review B, 56, 8997. https://doi.org/10.1103/PhysRevB.56.8997
[24]	Morris, J. (2015) New Scenarios for Classical and Quantum Mechanical Systems with Position Dependent Mass. Quantum Studies: Mathematics and Foundations, 2, 359-370. https://doi.org/10.1007/s40509-015-0037-7
[25]	Dekar, L., Chetouani, L. and Hammann, T. (1998) An Exactly Soluble Schrodinger Equation with Smooth Position-Dependent Mass. Journal of Mathematical Physics, 39, 2551. https://doi.org/10.1063/1.532407
[26]	Plastino, A., Puente, A., Casas, M., Garcias, F. and Plastino, A. (2000) Bound States in Quantum Systems with Position Dependent E ective Masses. Revista Mexicana de Fsica, 46, 78.
[27]	Alhaidari, A. (2002) Solutions of the Nonrelativistic Wave Equation with Position-Dependent E ective Mass. Physical Review A, 66, Article ID: 042116. https://doi.org/10.1103/PhysRevA.66.042116
[28]	Yu, J. and Dong, S. (2004) Exactly Solvable Potentials for the Schrodinger Equation with Spatially Dependent Mass. Physics Letters A, 235, 194-198. https://doi.org/10.1016/j.physleta.2004.03.056
[29]	Mustafa, O. and Mazharimousavi, S. (2009) Spherical-Separability of Non-Hermitian Hamiltonians and Pseudo-PTSymmetry. International Journal of Theoretical Physics, 48, 183-193. https://doi.org/10.1007/s10773-008-9794-y
[30]	Bagchi, B., Banerjee, A., Quesne, C. and Tkachuk, V. (2005) Deformed Shape Invariance and Exactly Solvable Hamiltonians with Position-Dependent E ective Mass. Journal of Physics A: Mathematical and General, 38, 2929. https://doi.org/10.1088/0305-4470/38/13/008
[31]	de Souza Dutra, A. and Almeida, C. (2000) Exact Solvability of Potentials with Spatially Dependent E ective Masses. Physics Letters A, 275, 25-30. https://doi.org/10.1016/S0375-9601(00)00533-8
[32]	Bagchi, B., Gorain, P., Quesne, C. and Roychoudhury, R. (2004) A General Scheme for the E ective-Mass Schrodinger Equation and the Generation of the Associated Potentials. Modern Physics Letters A, 19, 2765-2775. https://doi.org/10.1142/S0217732304016123
[33]	Levy-Leblond, J. (1995) Position-Dependent E ective Mass and Galilean Invariance. Physical Review A, 52, 1845. https://doi.org/10.1103/PhysRevA.52.1845
[34]	Chetouani, L., Dekar, L. and Hammann, T. (1995) Greens Functions via Path Integrals for Systems with Position-Dependent Masses. Physical Review A, 52, 82. https://doi.org/10.1103/PhysRevA.52.82
[35]	Yung, K. and Yee, J. (1994) Derivation of the Modi ed Schrodinger Equation for a Particle with a Spatially Varying Mass through Path Integrals. Physical Review A, 50, 104. https://doi.org/10.1103/PhysRevA.50.104
[36]	Rajbongshi, H. (2018) Exact Analytic Solution of Position-Dependent Mass Schrodinger Equation. Indian Journal of Physics, 92, 357-367. https://doi.org/10.1007/s12648-017-1108-x
[37]	Shewell, J. (1959) On the Formation of Quantum-Mechanical Operators. American Journal of Physics, 27, 16. https://doi.org/10.1119/1.1934740
[38]	Trabelsi A., Madouri F., Merdaci A. and Almatar A. (2013) Classi cation Scheme for Kinetic Energy Operators with Position-Dependent Mass. e-print arXiv: 1302.3963v1
[39]	Bender, C. and Milton, K. (1997) Nonperturbative Calculation of Symmetry Breaking in Quantum Field Theory. Physical Review D, 55, R3255(R). https://doi.org/10.1103/PhysRevD.55.R3255
[40]	Bender, C. and Boettcher, S. (1998) Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry. Physical Review Letters, 80, 5243. https://doi.org/10.1103/PhysRevLett.80.5243
[41]	Bender, C., Boettcher, S. and Meisinger, P. (1999) PT-Symmetric Quantum Mechanics. Journal of Mathematical Physics, 40, 2201. https://doi.org/10.1063/1.532860
[42]	Bender, C. (1999) The Complex Pendulum. Physics Reports, 315,27-40. https://doi.org/10.1016/S0370-1573(99)00024-1
[43]	Bender, C., Dunne, G.V. and Meisenger, P.N. (1999) Complex Periodic Potentials with Real Band Spectra. Physics Letters A, 252, 272-276. https://doi.org/10.1016/S0375-9601(98)00960-8
[44]	Bender, C. and Dunne, G.V. (1999) Large-Order Perturbation Theory for a Non-Hermitian PT-Symmetric Hamiltonian. Journal of Mathematical Physics, 40, 4616. https://doi.org/10.1063/1.532991
[45]	Bender, C., Boettcher, S. and Savage, V.M. (2000) Conjecture on the Interlacing of Zeros in Complex Sturm-Liouville Problems. Journal of Mathematical Physics, 41, 6381. https://doi.org/10.1063/1.1288247
[46]	Bender, C., Dunne, G., Meisenger, P. and Simsek, M. (2001) Quantum Complex H enon-Heiles Potentials. Physics Letters A, 281, 311-316. https://doi.org/10.1016/S0375-9601(01)00146-3
[47]	Bender, C., Berry, M., Meisenger, P., Savage, V. and Simsek, M. (2001) Complex WKB Analysis of Energy-Level Degeneracies of Non-Hermitian Hamiltonians. Journal of Physics A: Mathematical and General, 34, L31. https://doi.org/10.1088/0305-4470/34/6/101
[48]	Chargui, Y., Dhahbi, A. and Trabelsi, A., A Novel Approach for Constructing Kinetic Energy Operators with Position Dependent Mass. Submitted for publication.
[49]	Hassanabadi, H., Chung, W.S., Zare, S. and Alimohammadi, M. (2017) Scattering of Position-Dependent Mass Schrodinger Equation with Delta Potential. The European Physical Journal Plus, 132, 135. https://doi.org/10.1140/epjp/i2017-11422-0
[50]	Abramowitz, M. and Stegun, I.A. (1972) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. U.S. Government Printing O ce.
[51]	Carinena, J.F., Ranada, M.F. and Santander, M. (2007) A Quantum Exactly Solvable Non-Linear Oscillator with Quasi-Harmonic Behaviour. Annals of Physics, 322, 434-459. https://doi.org/10.1016/j.aop.2006.03.005
[52]	Midya, B. and Roy, B. (2009) A Generalized Quantum Nonlinear Oscillator. Journal of Physics A: Mathematical and Theoretical, 42, Article ID: 285301. https://doi.org/10.1088/1751-8113/42/28/285301
[53]	Mostafazadeh, A. (2002) Pseudo-Hermiticity versus PT Symmetry: The Necessary Condition for the Reality of the Spectrum of a Non-Hermitian Hamiltonian. Journal of Mathematical Physics, 43, 205. https://doi.org/10.1063/1.1418246
[54]	Mostafazadeh, A. (2002) Pseudo-Hermiticity versus PTSymmetry. II. A Complete Characterization of Non-Hermitian Hamiltonians with a Real Spectrum. Journal of Mathematical Physics, 43, 2814. https://doi.org/10.1063/1.1461427
[55]	Mostafazadeh, A. (2002) Pseudo-Hermiticity versus PTSymmetry III: Equivalence of Pseudo-Hermiticity and the Presence of Antilinear Symmetries. Journal of Mathematical Physics, 43, 3944. https://doi.org/10.1063/1.1489072
[56]	Kretschmer, R. and Szymanowski, L. (2004) Quasi-Hermiticity in In nite-Dimensional Hilbert Spaces. Physics Letters A, 325, 112-117. https://doi.org/10.1016/j.physleta.2004.03.044
[57]	Cooper, F., Khare, A. and Sukhatme, U. (1995) Supersymmetry and Quantum Mechanics. Physics Reports, 251, 267-385. https://doi.org/10.1016/0370-1573(94)00080-M
[58]	Zhao, F.Q., Liang, X.X. and Ban, S.L. (2003) In uence of the Spatially Dependent E ective Mass on Bound Polarons in Finite Parabolic Quantum Wells. The European Physical Journal B, 33, 3-8. https://doi.org/10.1140/epjb/e2003-00134-3
[59]	Mathews, P.M. and Lakshmanan, M. (1975) A Quantum- Mechanically Solvable Nonpolynomial Lagrangian with Velocity-Dependent Interaction. Il Nuovo Cimento A, 26, 299-316. https://doi.org/10.1007/BF02769015
[60]	Karthiga, S., Chithiika Ruby, V., Senthilvelan, M. and Lakshmanan, M. (2017) Quantum Solvability of a General Ordered Position Dependent Mass System: Mathews-Lakshmanan Oscillator. Journal of Mathematical Physics, 58, Article ID: 102110. https://doi.org/10.1063/1.5008993
[61]	Gonul, B., Gonul, B., Tutcu, D. and  Ozer, O. (2002) Supersymmetric Approach to Exactly Solvable Systems with Position-Dependent E ective Masses. Modern Physics Letters A, 17, 2057-2066. https://doi.org/10.1142/S0217732302008563
[62]	Arda, A. and Sever, R. (2011) Bound State Solutions of Schrodinger Equation for Generalized Morse Potential with Position-Dependent Mass. Communications in Theoretical Physics, 56, 51. https://doi.org/10.1088/0253-6102/56/1/09
[63]	Chen, Y., Yan, Z., Mihalache, D. and Malomed, B.A. (2017) Families of Stable Solitons and Excitations in the PT-Symmetric Nonlinear Schrodinger Equations with Position-Dependent E ective Masses. Scienti c Reports, 7, Article No. 1257. https://doi.org/10.1038/s41598-017-01401-3
[64]	Xie, Q.-T. (2012) New Quasi-Exactly Solvable Double-Well Potentials. Journal of Physics A: Mathematical and Theoretical, 45, Article ID: 175302. https://doi.org/10.1088/1751-8113/45/17/175302