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


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.

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Dhahbi, A. , Chargui, Y. and Trablesi, A. (2019) A New Class of Exactly Solvable Models within the Schrödinger Equation with Position Dependent Mass. Journal of Applied Mathematics and Physics, 7, 1013-1026. doi: 10.4236/jamp.2019.75068.

Conflicts of Interest

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


[1] Geller, M. and Kohn, W. (1993) Quantum Mechanics of Electrons in Crystals with Graded Composition. Physical Review Letters, 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.
[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.
[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).
[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.
[7] BenDaniel, D. and Duke, C. (1966) Space-Charge E ects on Electron Tunneling. Physical Review, 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.
[9] Bastard, G. (1981) Superlattice Band Structure in the Envelope-Function Approximation. Physical Review B, 24, 5693.
[10] Gora, T. and Williams, F. (1969) Theory of Electronic States and Transport in Graded Mixed Semiconductors. Physical Review, 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.
[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.
[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.
[15] Von Roos, O. (1985) Position-Dependent E ective Masses in Semiconductor Theory. II. Physical Review B, 31, 2294.
[16] Morrow, R. (1987) Establishment of an E ective-Mass Hamiltonian for Abrupt Heterojunctions. Physical Review B, 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.
[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.
[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.
[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.
[22] Foulkes, W. and Schluter, M. (1990) Pseudopotentials with Position-Dependent Electron Masses. Physical Review B, 42, Article ID: 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.
[24] Morris, J. (2015) New Scenarios for Classical and Quantum Mechanical Systems with Position Dependent Mass. Quantum Studies: Mathematics and Foundations, 2, 359-370.
[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.
[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.
[28] Yu, J. and Dong, S. (2004) Exactly Solvable Potentials for the Schrodinger Equation with Spatially Dependent Mass. Physics Letters A, 235, 194-198.
[29] Mustafa, O. and Mazharimousavi, S. (2009) Spherical-Separability of Non-Hermitian Hamiltonians and Pseudo-PTSymmetry. International Journal of Theoretical Physics, 48, 183-193.
[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.
[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.
[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.
[33] Levy-Leblond, J. (1995) Position-Dependent E ective Mass and Galilean Invariance. Physical Review A, 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.
[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.
[36] Rajbongshi, H. (2018) Exact Analytic Solution of Position-Dependent Mass Schrodinger Equation. Indian Journal of Physics, 92, 357-367.
[37] Shewell, J. (1959) On the Formation of Quantum-Mechanical Operators. American Journal of Physics, 27, 16.
[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).
[40] Bender, C. and Boettcher, S. (1998) Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry. Physical Review Letters, 80, 5243.
[41] Bender, C., Boettcher, S. and Meisinger, P. (1999) PT-Symmetric Quantum Mechanics. Journal of Mathematical Physics, 40, 2201.
[42] Bender, C. (1999) The Complex Pendulum. Physics Reports, 315,27-40.
[43] Bender, C., Dunne, G.V. and Meisenger, P.N. (1999) Complex Periodic Potentials with Real Band Spectra. Physics Letters A, 252, 272-276.
[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.
[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.
[46] Bender, C., Dunne, G., Meisenger, P. and Simsek, M. (2001) Quantum Complex H enon-Heiles Potentials. Physics Letters A, 281, 311-316.
[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.
[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.
[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.
[52] Midya, B. and Roy, B. (2009) A Generalized Quantum Nonlinear Oscillator. Journal of Physics A: Mathematical and Theoretical, 42, Article ID: 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.
[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.
[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.
[56] Kretschmer, R. and Szymanowski, L. (2004) Quasi-Hermiticity in In nite-Dimensional Hilbert Spaces. Physics Letters A, 325, 112-117.
[57] Cooper, F., Khare, A. and Sukhatme, U. (1995) Supersymmetry and Quantum Mechanics. Physics Reports, 251, 267-385.
[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.
[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.
[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.
[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.
[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.
[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.
[64] Xie, Q.-T. (2012) New Quasi-Exactly Solvable Double-Well Potentials. Journal of Physics A: Mathematical and Theoretical, 45, Article ID: 175302.

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