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Current issue   Ukr. J. Phys. 2016, Vol. 61, N 4, p.331-341
doi:10.15407/ujpe61.04.0331    Paper

Los V.F.1, Los M.V.2

1 Institute for Magnetism, Nat. Acad. Sci. of Ukraine
(36-b, Vernadsky Blvd., Kyiv 142, Ukraine)
2 Luxoft Eastern Europe
(14, Vasylkivs’ka Str., B, Business Center STEND, Kyiv 03040, Ukraine)

An Exact Solution of the Time-Dependent Schrödinger Equation with a Rectangular Potential for Real and Imaginary Times

Section: General Problems of Theoretical Physics
Original Author's Text: English

Abstract: A propagator for the one-dimensional time-dependent Schrödinger equation with an asymmetric rectangular potential is obtained, by using the multiple-scattering theory. It allows the consideration of the reflection and transmission processes as the scattering of a particle at the potential jump (in contrast to the conventional wave-like picture) and the account for the nonclassical counterintuitive contribution of the backward-moving component of the wave packet attributed to a particle. This propagator completely resolves the corresponding time-dependent Schrödinger equation (defines the wave function ψ(x,t)) and allows the consideration of the quantum mechanical effects of a particle reflection from the potential downward step/well and a particle tunneling through the potential barrier as a function of the time. These results are related to fundamental issues such as measuring the time in quantum mechanics (tunneling time, time of arrival, dwell time). For the imaginary time, which represents an inverse temperature (t → iħ/kBt), the obtained propagator is equivalent to the density matrix for a particle that is in a heat bath and is subject to the action of a rectangular potential. This density matrix provides information about particles’ density in the different spatial areas relative to the potential location and on the quantum coherence of different particle spatial states. If one passes to the imaginary time (t → it), the matrix element of the calculated propagator in the spatial basis provides a solution to the diffusion-like equation with a rectangular potential. The obtained exact results are presented as the integrals of elementary functions and thus allow a numerical visualization of the probability density |ψ(x,t)|2, the density matrix, and the solution of the diffusion-like equation. The results obtained may also be applied to spintronics due to the fact that the asymmetric (spin-dependent) rectangular potential can model the potential profile in layered magnetic nanostructures.

Key words: Schrödinger equation, asymmetric rectangular potential, layered magnetic nanostructures.

1. A.D. Baute, I.L. Egusquiza, and J.G. Muga, J. Phys. A: Math. Theor. 34, 42892001 (2001).
2. A.D. Baute, I.L. Egusquiza, and J.G. Muga, Int. J. Theor. Physics, Group Theory, and Nonlinear Optics 8, 1 (2002); (e-print arXiv: quant-ph/0007079).
3. Time in Quantum Mechanics, vol. 1 (Lecture Notes in Physics, vol. 734), edited by J.G. Muga, R. Sala Mayato, and I.L. Egusquiza (Springer, Berlin, 2008).
4. Time in Quantum Mechanics, vol. 2 (Lecture Notes in Physics, vol. 789), edited by J.G. Muga, A. Ruschhaup, and A. del Campo (Springer, Berlin, 2009).
5. M.N. Baibich, J.M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas, Phys. Rev. Lett. 61, 2472 (1988).   CrossRef   PubMed
6. R. Julliere, Phys. Lett. A 54, 225 (1975);   CrossRef
P. LeClair, J.S. Moodera, and R. Meservay, J. Appl. Phys. 76, 6546 (1994).   CrossRef
7. R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
8. M. Kac, Probability and Related Topics in Physical Sciencies, Lectures in Applied Mathematics, vol. 1 (Interscience, London, New York, 1958).
9. A.O. Barut and I.H. Duru, Phys. Rev. A 38, 5906 (1988).   CrossRef
10. L.S. Schulman, Phys. Rev. Lett.,49, 599 (1982).   CrossRef
11. T.O. de Carvalho, Phys. Rev. A 47, 2562 (1993).   CrossRef   PubMed
12. J.M. Yearsley, J. Phys. A: Math. Theor. 41, 285301 (2008).   CrossRef
13. V.F. Los and A.V. Los, J. Phys. A: Math. Theor. 43, 055304 (2010).   CrossRef
14. V.F. Los and A.V. Los, J. Phys. A: Math. Theor. 44, 215301 (2011).   CrossRef
15. V.F. Los and M.V. Los, J. Phys. A: Math. Theor. 45, 095302 (2012).   CrossRef
16. J.G. Muga and C.R. Leavens, Phys. Rep. 338, (2000).
17. V.F. Los and M.V. Los, Theor. Math. Phys. 177, 1704 (2013).   CrossRef
18. E.N. Economou, Green's Functions in Quantum Physics (Springer, Berlin, 1979).   CrossRef   PubMed