IDs:

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

**PubDateOther :**

**Volume :**

**Series :**

**StartPage :**
59

**EndPage :**
72

**Publisher :**

**Address :**

**ISBN :**
80-7040-130-3

**Keywords :**
TOP_Misc;TOP_Tunnel;YEAR_Before1998;W_Physedu;W_Schroed;W_WP;W_Theory

**Notes :**

**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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