E101 - Institut für Analysis und Scientific Computing
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Date (published):
2013
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Number of Pages:
85
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Keywords:
Numerische Mathematik; Schrödingergleichung
de
Numerical methods; Schrödinger equation
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Abstract:
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de
The desire for understanding the world around us has been a drive for the development of different scientific disciplines like physics, biology, chemistry, mathematics. The most common approach for supporting a theory about a given physical problem is to create an appropriate experiment in the laboratory. Owing to mathematical modeling computers have become a commonly used kind of laboratories in the modern age.<br />Mathematical modeling is based on differential equations and gives a simulation of behavior of a system from any scientific discipline.<br />Quantum mechanics is a branch of physics that aimes to describe the microscopic world. Considered problems are for example modeling of an electron or an interaction of molecules. The main equation of this theory is the Schrödinger equation. Its solutions are in the form of a wave that describes the probability of finding a particle at a special place. This equation does not have an explicit solution for every case. For this reason the solution is approximated by numerical methods.<br />The first chapter of this work is dedicated to the introduction of the linear Schrödinger equation. The most frequent mathematical tools to prove the existence of the solutions in the case of the free particle model and the oscillatory potential model are presented. An explicit form of the solutions in these special cases is also given.<br />In the next chapters (2,3,4) numerical methods applied to the Schrödinger equation are presented. These methods are finite difference methods, spectral methods and the Visscher method. The numerical method should give a convergent approximation to the solution, should be stable in time and preserve the physical features of the solution. Specifically, as the exact solution that describes a particle has constant mass and energy, the numerical solution should also preserve these quantities.<br />To understand the qualities of these numerical methods applied to the Schrödinger equation, some proofs are presented first. These methods are also implemented in the programing language MATLAB and the numerical solution of the free particle model and the oscillator potential model are compared to the exact solution which is already known from chapter 1. In this way the characteristics of the numerical methods are confirmed and summarized in the conclusion (table 5.1) at the end of this work
