Schrödinger Equation with Position-Dependent Mass: Staggered Mass Distributions

Authors: J. J. Peña, J. Morales, J. García-Ravelo, L. Arcos-Díaz

Abstract:

The Point canonical transformation method is applied for solving the Schrödinger equation with position-dependent mass. This class of problem has been solved for continuous mass distributions. In this work, a staggered mass distribution for the case of a free particle in an infinite square well potential has been proposed. The continuity conditions as well as normalization for the wave function are also considered. The proposal can be used for dealing with other kind of staggered mass distributions in the Schrödinger equation with different quantum potentials.

Keywords: Free particle, point canonical transformation method, position-dependent mass, staggered mass distribution.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1131401

References:

[1] O. Von Roos, “Position-dependent effective mass in semiconductor theory”, Phys. Rev. B 27, 7547 (1983).
[2] D. L. Smith, C. Mailhiot, “Theory of semiconductor superlattice electronic structure”, Rev. Mod. Phys. 62, 173 (1990).
[3] M. Pi, S. M. Gatica, E. S. Hernandez and J. Navarro, “Structure and energetics of mixed 4He-3He drops”, Phys. Rev. B 56, 8997 (1997).
[4] G. T. Einevoll, “Operator ordering in effective mass theory for heterostructures II. Strained systems”, Phys. Rev. B 42, 3497 (1990).
[5] R. A. Morrow, “Establishment of effective-mass Hamiltonian for abrupt heterojunctions”, Phys. Rev. B 35, 8074 (1987).
[6] J. J. Peña, J. Morales, E. Zamora-Gallardo, J. García-Ravelo, “Isospectral Orthogonal Polynomials from the Darboux Transforms” Int. Journ. Quant. Chem., 100, 957 (2004).
[7] J. J. Peña, G. Ovando., J. Morales, J. García-Ravelo, J. Garcia, “Solvable Quantum Potentials with Special Functions Solutions”, Int. J. Quantum Chem Vol.108, pp. 1750, (2008).
[8] C. Quesne, V. M. Tkachuk, “Deformed algebras, position-dependent effective masses and curved spaces: an exactly solvable Coulomb problem”, J. Phys. A: Math. Gen., 37, 4267, (2004).
[9] Z. D. Chen, G. Chen, “Series solutions of the N-dimensional effective-mass Schrödinger equation with the power law potential”, Physica Scripta 72, 11 (2005).
[10] B. Bagchi, P. Gorain, C. Quesne, R. Roychoudhury, “A general scheme for the effective-mass Schrödinger equation and the generation of the associated potentials”, Mod. Phys. Lett. A 19, 2765 (2004).
[11] B. Gonul, B. Gonul, D. Tutco, O. Ozer, ”Supersymmetric approach to exactly solvable systems with position-dependent effective mass”, Mod. Phys. Lett. A, Vol. 17, pp. 2057, (2002).
[12] A. Schulze-Halberg, “Darboux transformations for time-dependent Schrödinger equations with effective mass”, Int. J. Mod. Phys. A. Vol. 21, pp. 1359, (2006).
[13] J. Garcia-Martinez, J. Garcia-Ravelo, A. Schulze-Halberg, J. J Peña, “Exactly solvable energy-dependent potentials”, Physics Letters A, Vol. 373, pp. 3619, (2009).
[14] J. J. Peña, G. Ovando., J. Morales, J. García-Ravelo, C. Pacheco-García, “Exactly solvable Schrodinger equation whit a position-dependent mass: Null Potential”, Int. J. Quantum Chem Vol.107, pp. 3039, (2007). J. Yu, S. H. Dong, Phys. Lett. A 325, 194 (2004).
[15] A. R. Plastino, A. Rigo, M. Casas, F. Garcias, and A. Plastino, “Supersymmetric approach to quantum systems with position-dependent effective mass. Phys. Rev. A., Vol. 60, pp. 4318, (1999).
[16] R. N. Costa-Filho, M. P. Almeida, G. A. Farias and J. S. Andrade Jr. “Displacement operator for quantum systems with position-dependent mass”, Phys. Rev. A, 84, 050102 (2011).
[17] J. M. Lèvy-Leblon “Elementary quantum models with position-dependent mass”, Eur. J. Phys. Vol. 13, pp 215-218, (1992).
[18] Qi-Gao Zhu and H. Kroemer, “Interface connection rules for effective-mass wave functions at an abrupt heterojunction between two different semiconductors”, Phys. Rev. B 27, 3519 (1983).
[19] G. Bastards, “Superlattice band structure in the envelop-function approximation”, Phys. Rev. B 24,10, 5693 (1981).
[20] Dekar L., L. Chetouani and T. F. Hammann, “Wave functions for smooth potential and mass step”. Phys. Rev. A., Vol. 59, pp. 107, (1999).
[21] R. Renan, J. M. Pereira, J. Riveiro, V. N. Freire, G. A. Farias, “The effect of a position dependent effective mass on the transmission of electrons through a double graded barrier”, Brazilian Journal of Physics, Vol. 24, No. 1, pp. 192, March, (1994).