Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access December 1, 2005

Quantum systems with effective and time-dependent masses: form-preserving transformations and reality conditions

  • Axel Schulze-Halberg EMAIL logo
From the journal Open Physics

Abstract

We study the time-dependent Schrödinger equation (TDSE) with an effective (position-dependent) mass, relevant in the context of transport phenomena in semiconductors. The most general form-preserving transformation between two TDSEs with different effective masses is derived. A condition guaranteeing the reality of the potential in the transformed TDSE is obtained. To ensure maximal generality, the mass in the TDSE is allowed to depend on time also.

[1] T. Gora and F. Williams: “Electronic states of homogeneous and inhomogeneous mixed semiconductors”, In: D.G. Thomas (Ed.):II–VI Semiconducting Compounds, Benjamin, New York, 1967. Search in Google Scholar

[2] T. Gora and F. Williams: “Theory of electronic states and transport in graded mixed semiconductors”, Phys. Rev., Vol. 177, (1969), pp. 1179–1182. http://dx.doi.org/10.1103/PhysRev.177.117910.1103/PhysRev.177.1179Search in Google Scholar

[3] G.T. Landsberg:Solid state theory: methods and applications, Wiley-Interscience, London, 1969. Search in Google Scholar

[4] O. von Roos: “Position-dependent effective masses in semiconductor theory”, Phys. Rev. B, Vol. 27, (1983), pp. 7547–7552. http://dx.doi.org/10.1103/PhysRevB.27.754710.1103/PhysRevB.27.7547Search in Google Scholar

[5] O. von Roos and H. Mavromatis: “Position-dependent effective masses in semiconductor theory. II”, Phys. Rev. B, Vol. 31, (1985), pp. 2294–2298. http://dx.doi.org/10.1103/PhysRevB.31.229410.1103/PhysRevB.31.2294Search in Google Scholar

[6] J.-M. Lévy-Leblond: “Position-dependent effective mass and Galilean invariance”, Phys. Rev. A, Vol. 52(3), (1995), pp. 1845–1849. http://dx.doi.org/10.1103/PhysRevA.52.184510.1103/PhysRevA.52.1845Search in Google Scholar

[7] L. Dekar, L. Chetouani and T.F. Hammann: “An exactly soluble Schrödinger equation with smooth position-dependent mass”, J. Math. Phys., Vol. 39, (1998), pp. 2551–2563. http://dx.doi.org/10.1063/1.53240710.1063/1.532407Search in Google Scholar

[8] Á. de Souza Dutra and C.A.S. Almeida: “Exact solvability of potentials with spatially dependent effective masses”, Phys. Lett. A, Vol. 275, (2000), pp. 25–30. http://dx.doi.org/10.1016/S0375-9601(00)00533-810.1016/S0375-9601(00)00533-8Search in Google Scholar

[9] A.D. Alhaidari: “Solutions of the nonrelativistic wave equation with position-dependent effective mass”, Phys. Rev. A, Vol. 66, (2002), pp. 042116. http://dx.doi.org/10.1103/PhysRevA.66.04211610.1103/PhysRevA.66.042116Search in Google Scholar

[10] B. Gönül, B. Gönül, D. Tutcu and O. Özer: “Supersymmetric approach to exactly solvable systems with position-dependent effective masses”, Modern Phys. Lett. A, Vol. 17, (2002), pp. 2057–2066. http://dx.doi.org/10.1142/S021773230200856310.1142/S0217732302008563Search in Google Scholar

[11] R. Koç and M. Koca: “A systematic study on the exact solution of the position dependent mass Schrödinger equation”, J. Phys. A, Vol. 36, (2003), pp. 8105–8112. http://dx.doi.org/10.1088/0305-4470/36/29/31510.1088/0305-4470/36/29/315Search in Google Scholar

[12] J. Yu. Dong, S.-H. Dong and G.-H. Sun: “Series solutions of the Schrödinger equation with position-dependent mass for the Morse potential”, Phys. Lett. A., Vol. 322, (2004), pp. 290–297. http://dx.doi.org/10.1016/j.physleta.2004.01.03910.1016/j.physleta.2004.01.039Search in Google Scholar

[13] C. Quesne and V.M. Tkachuk: “Deformed algebras, position-dependent effective masses and curved spaces: an exactly solvable Coulomb problem”, J. Phys. A, Vol. 37, (2004), pp. 4267–4281. http://dx.doi.org/10.1088/0305-4470/37/14/00610.1088/0305-4470/37/14/006Search in Google Scholar

[14] Y.C. Ou, Z. Cao and Q. Shen: “Energy eigenvalues for the systems with position-dependent effective mass”, J. Phys. A, Vol. 37, (2004), pp. 4283–4288. http://dx.doi.org/10.1088/0305-4470/37/14/00710.1088/0305-4470/37/14/007Search in Google Scholar

[15] G. Chen and Z.-D. Chen: “Exact solutions of the position-dependent Schrödinger equation in D dimensions”, Phys. Lett. A, Vol. 331, (2004), pp. 312–315. http://dx.doi.org/10.1016/j.physleta.2004.09.01210.1016/j.physleta.2004.09.012Search in Google Scholar

[16] A. Jannussis, G. Karayannis, P. Panagopoulos, V. Papatheou, M. Symeonidis, D. Vavougios, P. Siafarikas and V. Zisis: “Exactly soluble harmonic oscillator for a particular form of time and coordinates-dependent mass”, J. Phys. Soc. Japan, Vol. 53, (1984), pp. 957–962. http://dx.doi.org/10.1143/JPSJ.53.95710.1143/JPSJ.53.957Search in Google Scholar

[17] A. Schulze-Halberg: “Form-preserving transformations of the time-dependent Schrödinger equation with time- and position-dependent mass”, Commun. Theor. Phys. (Beijing), Vol. 43, (2005), pp. 657–665. http://dx.doi.org/10.1088/0253-6102/43/4/01710.1088/0253-6102/43/4/017Search in Google Scholar

[18] F. Finkel, A. Gonzalez-Lopez, N. Kamran and M.A. Rodriguez: “On form-preserving transformations for the time-dependent Schrodinger equation”, J. Math. Phys., Vol. 40, (1999), pp. 3268–3274. http://dx.doi.org/10.1063/1.53288510.1063/1.532885Search in Google Scholar

[19] M. Znojil (Ed.): “Proceedings of the 1st international workshop: pseudo-hermitian Hamiltonians in quantum physics”, Czech. J. Phys., Vol. 54, (2004), pp. 1–156. Search in Google Scholar

[20] Á.de Souza Dutra, M.B. Hott and V.G.C.S. dos Santos: “Non-Hermitian time-dependent quantum systems with real energies”, quant-ph/0311044. Search in Google Scholar

[21] G.T. Einevoll and P.C. Hemmer: “The effective-mass Hamiltonian for abrupt heterostructures”, J. Phys. C, Vol. 21, (1988), pp. L1193-L1198. http://dx.doi.org/10.1088/0022-3719/21/36/00110.1088/0022-3719/21/36/001Search in Google Scholar

[22] G.T. Einevoll, P.C. Hemmer and J. Thomsen: “Operator ordering in effectivemass theory for heterostructures. I. Comparison with exact results for superlattices, quantum wells, and localized potentials”, Phys. Rev. B, Vol. 42, (1990), pp. 3485–3496. http://dx.doi.org/10.1103/PhysRevB.42.348510.1103/PhysRevB.42.3485Search in Google Scholar

[23] G.T. Einevoll: “Operator ordering in effective-mass theory for heterostructures. II. Strained systems”, Phys. Rev. B, Vol. 42, (1990), pp. 3497–3502. http://dx.doi.org/10.1103/PhysRevB.42.349710.1103/PhysRevB.42.3497Search in Google Scholar

[24] R.A. Morrow and K.R. Brownstein: “Model effective-mass Hamiltonians for abrupt heterojunctions and the associated wave-function-matching conditions”, Phys. Rev. B, Vol. 30, (1984), pp. 678–680. http://dx.doi.org/10.1103/PhysRevB.30.67810.1103/PhysRevB.30.678Search in Google Scholar

[25] R.A. Morrow: “Establishment of an effective-mass Hamiltonian for abrupt heterojunctions”, Phys. Rev. B, Vol. 35, (1987), pp. 8074–8079. http://dx.doi.org/10.1103/PhysRevB.35.807410.1103/PhysRevB.35.8074Search in Google Scholar

[26] J. Thomsen, G.T. Einevoll and P.C. Hemmer: “Operator ordering in effective-mass theory”, Phys. Rev. B, Vol. 39, (1989), pp. 12783–12788. http://dx.doi.org/10.1103/PhysRevB.39.1278310.1103/PhysRevB.39.12783Search in Google Scholar PubMed

[27] Q.-G. Zhu and H. Kroemer: “Interface connection rules for effective-mass wave functions at an abrupt heterojunction between two different semiconductors”, Phys. Rev. B, Vol. 27, (1983), pp. 3519–3527. http://dx.doi.org/10.1103/PhysRevB.27.351910.1103/PhysRevB.27.3519Search in Google Scholar

[28] K.C. Yung and J.H. Yee: “Derivation of the modified Schrödinger equation for a particle with a spatially varying mass through path integrals”, Phys. Rev. A, Vol. 50, (1994), pp. 104–106. http://dx.doi.org/10.1103/PhysRevA.50.10410.1103/PhysRevA.50.104Search in Google Scholar

[29] E. Kamke:Differentialgleichungen—Lösungsmethoden und Lösungen, B.G. Teubner, Stuttgart, 1983. Search in Google Scholar

[30] L. Dekar, L. Chetouani and T.F. Hammann: “Wave function for smooth potential and mass step”, Phys. Rev. A, Vol. 59, (1999), pp. 107–112. http://dx.doi.org/10.1103/PhysRevA.59.10710.1103/PhysRevA.59.107Search in Google Scholar

Published Online: 2005-12-1
Published in Print: 2005-12-1

© 2005 Versita Warsaw

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 28.3.2024 from https://www.degruyter.com/document/doi/10.2478/BF02475615/html
Scroll to top button