We discuss a solution of the time-dependent Schrödinger equation that incorporates absorbing boundary conditions and a method for extracting the reflection and transmission probabilities for wave packets interacting with time-dependent potential barriers. We apply the method to a rectangular barrier that moves with constant velocity, an oscillating rectangular barrier, a locally periodic barrier with an amplitude modulated by a traveling wave, and a locally periodic potential with an amplitude modulated by a standing wave. Visualizations of the reflection phenomena are presented with an emphasis on understanding these systems from their dynamics. Applications to non-stationary neutron optics experiments are discussed briefly.
REFERENCES
1.
A.
Goldberg
, H. M.
Schey
, and J. L.
Schwartz
, “Computer-generated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena
,” Am. J. Phys.
35
(3
), 177
–186
(1967
).2.
D. L.
Haavig
and R.
Reifenberger
, “Dynamic transmission and reflection phenomena for a time-dependent rectangular potential
,” Phys. Rev. B
26
(12
), 6408
–6420
(1982
).3.
M. L.
Chiofalo
, M.
Artoni
, and G. C.
La Rocca
, “Atom resonant tunnelling through a moving barrier
,” New J. Phys.
5
, 78
–1
78
–15
(2003
).4.
M. R. A.
Shegelski
, T.
Poole
, and C.
Thompson
, “Capture of a quantum particle by a moving trapping potential
,” Eur. J. Phys.
34
, 569
–590
(2013
).5.
J.
Penfold
and R. K.
Thomas
, “The application of the specular reflection of neutrons to the study of surfaces and interfaces
,” J. Phys.: Condens. Matter
2
, 1369
–1412
(1990
).6.
J. F.
Ankner
, C. F.
Majkrzak
, and S. K.
Satija
, “Neutron reflectivity and grazing angle diffraction
,” J. Res. Natl. Inst. Stand. Technol.
98
(1
), 47
–58
(1993
).7.
J.
Lekner
, “Reflection of neutrons by periodic stratifications
,” Physica B
202
, 16
–22
(1994
).8.
E.
Raitman
, V.
Gavrilov
, and Ju.
Ekmanis
, “Neutron diffraction on acoustic waves in perfect and deformed single crystals
,” in Modeling and Measurement Methods for Acoustic Waves and for Acoustic Microdevices
, edited by Marco G.
Beghi
(InTech
, 2010
).9.
J.
Felber
, R.
Gähler
, and C.
Rausch
, “Matter waves at a vibrating surface: Transition from quantum-mechanical to classical behavior
,” Phys. Rev. A
53
(1
), 319
–328
(1996
).10.
M.
Utsuro
and V. K.
Ignatovich
, Handbook of Neutron Optics
(Wiley-VCH, Verlag GmbH & Co. KGaA
, 2010
).11.
A.
Steane
, P.
Szriftgiser
, P.
Desbiolles
, and J.
Dalibard
, “Phase modulation of atomic de Broglie waves
,” Phys. Rev. Lett.
74
(25
), 4974
–4975
(1995
).12.
13.
R. W.
Robinett
, Quantum Mechanics: Classical Results, Modern Systems, and Visualized Examples (Oxford U.P., New York & Oxford, 1997
).14.
R.
Gilmore
, Elementary Quantum Mechanics in One Dimension
(The Johns Hopkins University Press
, Baltimore & London
, 2004
).15.
M. H.
Bramhall
and B. M.
Casper
, “Reflections on a wave packet approach to quantum mechanical barrier penetration
,” Am. J. Phys.
38
(9
), 1136
–1145
(1970
).16.
A specific exception is for the case in which a barrier moves with constant velocity, vb. If the reflectivity R(k0) of the static barrier is known, then the reflectivity of the moving barrier is simply a shifted version of the static reflectivity, .
17.
P. B.
Visscher
, “A fast explicit algorithm for the time-dependent Schrödinger equation
,” Comput. Phys.
5
(6
), 596
–598
(1991
).18.
W.
Dai
, G.
Li
, R.
Nassar
, and S.
Su
, “On the stability of the FDTD method for solving a time-dependent Schrödinger equation
,” Numer. Methods Partial Differ. Equ.
21
(6
), 1140
–1154
(2005
).19.
We tested the accuracy of the algorithm by integrating the wave packet forward in time for N time steps, during which it interacted with a time-varying potential. After N time steps the wave function was replaced by its complex conjugate and then integrated for another N time steps forward in time (but the potential motion was reversed). The resulting wave packet moved backward in time as expected, eventually lying on top of the initial wave packet. The difference between the initial and final wave packet probability densities integrated over the full-width at half maximum was 0.005%.
20.
T.
Shibata
, “Absorbing boundary conditions for the finite-difference time-domain calculation of the one-dimensional Schrödinger equation
,” Phys. Rev. B
43
(8
), 6760
–6763
(1991
).21.
T.
Paul
, M.
Hartung
, K.
Richter
, and P.
Schlagheck
, “Nonlinear transport of Bose-Einstein condensates through mesoscopic waveguides
,” Phys. Rev. A
76
, 063605
–1
(2007
).22.
F. L.
Dubeibe
, “Solving the time-dependent Schrödinger equation with absorbing boundary conditions and source terms in Mathematica 6.0
,” Int. J. Mod. Phys. C
21
(11
), 1391
(2010
).23.
H.-W.
Lee
, A.
Zysnarski
, and P.
Kerr
, “One-dimensional scattering by a locally periodic potential
,” Am. J. Phys.
57
(8
), 729
–734
(1989
).24.
D. J.
Griffiths
and N. F.
Taussig
, “Scattering from a locally periodic potential
,” Am. J. Phys.
60
(10
), 883
–888
(1992
).25.
D. J.
Griffiths
and C. A.
Steinke
, “Waves in locally periodic media
,” Am. J. Phys.
69
(2
), 137
–154
(2001
).26.
Upon closer inspection, the peak location is actually closer to k0 = 606. This shift in the peak is due to dynamical effects as described in Ref. 7, Sec. 4.
27.
A.
Meyer
, R. M.
Dimeo
, P. M.
Gehring
, and D. A.
Neumann
, “The high-flux backscattering spectrometer at the NIST Center for Neutron Research
,” Rev. Sci. Instrum.
74
(5
), 2759
–2777
(2003
).28.
W. A.
Hamilton
, A. G.
Klein
, G. I.
Opat
, and P. A.
Timmins
, “Neutron diffraction by surface acoustic waves
,” Phys. Rev. Lett.
58
(26
), 2770
–2773
(1987
).29.
B.
Buras
and T.
Giebultowicz
, “Modified Ewald construction for neutrons reflected by moving lattices
,” Acta Crystallogr.
28
(2
), 151
–153
(1972
).© 2014 American Association of Physics Teachers.
2014
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