Solving the time-dependent Schrödinger equation is crucial for understanding the dynamics of quantum mechanical systems. Since the analytical solutions are very limited, it
is necessary to use numerical methods. In the thesis the approach with finite difference
methods is presented. Widely used is the simple Crank-Nicolson method. The thesis
presents its advantages and how it can be further improved. I use the presented method to
investigate the time evolution of a Gaussian wave packet that represents a localized state,
in a given potential.
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