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14

Authors : Márk,G.,I.;Pacher,Pál
Title : The study of time dependent scattering using Fourier transform technique
BookTitle : Proceedings, ICOMM'95, International conference on mathematical methods in science and technology, 3-6 June 1995, Vienna
Editors : Kainz,W.
PubDateYear : 1995
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StartPage : 59
EndPage : 72
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ISBN : 80-7040-130-3
Keywords : TOP_Misc;TOP_Tunnel;YEAR_Before1998;W_Physedu;W_Schroed;W_WP;W_Theory
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Abstract : To get information on the evolution of state during the scattering process one has to solve the quantummechanical equation of motion - the time-dependent Schr”dinger equation. For realistic systems this requires the numerical solution of the Schr”dinger equation in two or three dimensions which is a difficult and time consuming task using traditional methods. An efficient numerical technique - the split-operator Fourier transform method - helps to overcome the difficulties. The evolution operator is written as a product of three exponentials which contain the kinetic or potential energy operators; thus the evaluation of its action on the wave function is split into three steps. The evolution of the wavefunction over a time increment dt is approximated by the product of a free-particle evolution for one-half the time increment, a potential-only evolution for a full time increment, and a final free-particle evolution for another half time increment. Evaluation of the effect of the exponential containing the kinetic energy operator on the wave function utilises the property of Fourier transform that differentiation of a function in coordinate space is equivalent to multiplication of the function's representation in the Fourier transform space (k space) with the conjugate variable k. Fast Fourier transform (FFT) is used to perform the Fourier integrals. For a given potential the wave function is calculated at different time instants on a 512 * 512 grid of the coordinate space. The probability density function and the real part of the wave function are displayed in two dimensional colour graphs which can also be used for computer animation.
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