Dynamics of the probability density in the internal region of a double for a Dirac delta potential


  • Alberto Hernández Maldonado Faculty of Engineering Sciences and Technology, Autonomous University of Baja California, Valle de las Palmas Unit, Tijuana, Baja California, Mexico. https://orcid.org/0000-0002-9768-4060




Quantum transients, Double delta potential, Moshinsky shutter.


Based on exact analytical solutions of the time-dependent Schrödinger equation, and initial conditions of plane waves cut in x-space by an absorbing or reflecting quantum shutter, the dynamic behavior of the electronic probability density or buildup is analyzed, in the internal region of a double Dirac delta potential double, from very short times (given in half-lives), until reaching the stationary situation. The buildup is compared between both cases, absorbent and reflective quantum shutter, finding that for short times the buildup depends on the initial condition (IC) used, while for long times, it is independent of said IC. Using the reflecting IC, the buildup is analyzed from practically zero time, until the steady-state is reached. The results found may be relevant in the context of rapid response in devices known as resonant tunneling diodes.


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M. Moshinsky, “Diffraction in time”. Phys. Rev. Vol. 88, No. 1, 1952. https://doi.org/10.1103/PhysRev.88.625. DOI: https://doi.org/10.1103/PhysRev.88.625

Born, M. E. Wolf. “Principles of optics,” Pergamon Press, Inc. Sixty edition. New York, 1959, pp. 808. https://www.iaa.csic.es/~dani/ebooks/Optics/Principles%20of%20Optics%20-%20M.Born,%20E.%20Wolf.pdf

A. S. Gerasimov y M. V. Kazarnovskii, “Possibility of observing nontationary quantum-mechanical effects by means of ultracold neutron,” Sov. Phys. JETP, Vol. 44, pp. 892-987, 1976 http://jetp.ac.ru/cgi-bin/dn/e_044_05_0892.pdf.

R. Gähler, A. G. Klein y A. Zeilinger, “Neutron optical tests of nonlinear wave mechanics”, Phys. Rev. A, Vol. 23, pp. 1611-1617, 1981. https://doi.org/10.1103/PhysRevA.23.1611. DOI: https://doi.org/10.1103/PhysRevA.23.1611

J. Felber, R. Gähler, C. Raush y R. Golub, “Matter waves at a vibrating surface: Transition from quantum-mechanical to classical behavior”, Phys. Rev. A, Vol. 53, pp. 319-328, 1996. https://doi.org/10.1103/PhysRevA.53.319. DOI: https://doi.org/10.1103/PhysRevA.53.319

A. del Campo, G. García-Calderón y J.G. Muga, “Quantum transients” Phys. Rep. vol. 476, pp. 1-50, 2009. https://doi.org/10.1016/j.physrep.2009.03.002. DOI: https://doi.org/10.1016/j.physrep.2009.03.002

J. Villavicencio, R. Romo y A. Hernández-Maldonado, “Absorption dynamics and delay time in complex potentials”, Phys. Scr. Vol. 93, no. 055201, 2018. https://orcid.org/0000-0002-2523-6584. DOI: https://doi.org/10.1088/1402-4896/aab41c

P. Szriftgiser, D. Guërey-Odelin, M. Arndt, y J. Dablibard, “Atomic wave diffraction and interference using temporal slits”, Phys. Rev. Lett., vol. 77, no. 1, pp. 4-7, 1996. https://doi.org/10.1103/PhysRevLett.77.4. DOI: https://doi.org/10.1103/PhysRevLett.77.4

T. Hils, J. Felber, R. Gähler, W. Gläaser, R. Golub, K. Habicht, y P. Wille. “Matter-wave optics in the time domain: Results of a cold-neutron experiment”. Phys. Rev. A, Vol. 58, pp. 4784-4790, 1998. https://doi.org/10.1103/PhysRevA.58.4784. DOI: https://doi.org/10.1103/PhysRevA.58.4784

G. García-Calderón y A. Hernández-Maldonado, “Delay time as a quantum transient interference effect”, Physical Review A, Vol. 86, no. 062118, 2012. https://doi.org/10.1103/PhysRevA.86.062118. DOI: https://doi.org/10.1103/PhysRevA.86.062118

A. Hernández, “Delay time in a single barrier for a movable quantum shutter” Phys. Rev. A, Vol. 81, No. 054101, 2010. https://doi.org/10.1103/PhysRevA.81.054101. DOI: https://doi.org/10.1103/PhysRevA.81.054101

G. García-Calderón y A. Rubio. “Transient effects and delay time in the dynamics of resonant tunneling”. Phys Rev. A, Vol. 56, pp. 3361-3370, 1997. https://doi.org/10.1103/PhysRevA.55.3361. DOI: https://doi.org/10.1103/PhysRevA.55.3361

J. Villavicencio y R. Romo., “Dynamical analysis of the buildup process near resonance”, Appl. Phys. Lett., Vol. 77, no. 3, pp. 379-381, 2000. https://doi.org/10.1063/1.126982. DOI: https://doi.org/10.1063/1.126982

J. Villavicencio y A. Hernández-Maldonado, “Transient quantum beats, Rabi oscillations, and delay time of modulated matter waves”, Phys Rev. A, Vol. 101, no. 042109, 2020. https://doi.org/10.1103/PhysRevA.101.042109. DOI: https://doi.org/10.1103/PhysRevA.101.042109

R. Romo y J. Villavicencio. “Dynamical description of the buildup process in resonant tunneling: evidence of exponential and nonexponential contributions”. Phys. Rev. B Vol. 60: R2142-R2145, 1999. https://doi.org/10.1103/PhysRevB.60.R2142. DOI: https://doi.org/10.1103/PhysRevB.60.R2142

R. Romo y J. Villavicencio. “Role of the buildup oscillations on the speed of resonant tunneling diodes”. Appl. Phys. Lett. Vol. 78, pp. 1769-1771, 2001. https://doi.org/10.1063/1.1354164. DOI: https://doi.org/10.1063/1.1354164

R. Romo, J. Villavicencio y G. García-Calderón. “Transient tunneling effects of resonance doublets in triple barrier systems”. Phys. Rev. B, Vol. 66, no. 033108, 2002. https://doi.org/10.1103/PhysRevB.66.033108. DOI: https://doi.org/10.1103/PhysRevB.66.033108

A. Hernández Maldonado, “Estudio dinámico del retraso temporal en sistemas cuánticos”, Tesis. (CICESE), pp. 119, 2004.

A. Hernández Maldonado, R. Romo y J. Villavicencio, “Convergence of resonance expansions in quantum wave buildup”, Revista Mexicana de Física, Vol. 62, pp 257–266, 2016. http://www.scielo.org.mx/scielo.php?script=sci_arttext&pid=S0035-001X2016000300257.

Buildup or probability density at different times, using the reflective shutter initial condition.



How to Cite

Hernández Maldonado, A. (2020). Dynamics of the probability density in the internal region of a double for a Dirac delta potential. REVISTA DE CIENCIAS TECNOLÓGICAS, 3(3), 173–180. https://doi.org/10.37636/recit.v33173180



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