Applied Quantum and Statistical Physics

Eigenfunction Expansion Tutorial

Abstract

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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