This document describes how, given an initial wave function, the wave function at time t can be found. First the initial wave function is decomposed into an expansion of the Hamiltonian eigenfunctions. Time dependence is then applied to these eigenfunctions. Finally, the wave function in x-space is recreated from the expansion.
A free particle gaussian wave packet is used as an example. All calculations are done discretely, such that they may readily be implemented in MATLAB®.
Documents are available below as PDF files.
- Notation (PDF)
- Free Particle Time Dependence (PDF)
- Computing the Fourier Transform (PDF)
- Computing the time dependent amplitude function (PDF)
- Computing the Inverse Fourier Transform (PDF)
- Numerical Considerations (PDF)
- Generalizations: Adding an extra parameter (PDF)
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