Quantum Delocalization and Revival

Live Script exploring the time-dependent Schrödinger equation in one dimension and wave function delocalization and revival.

The time development of free particle wave solutions to the Schrödinger equation differs from that of acoustic and light waves. A free acoustic or light wave packet travels without change of shape while a free particle quantum wave packet may expand or contract. The difference can be understood as a consequence of the fact that the phase velocity of the quantum component waves depends upon their frequency.

When a localized wave packet reflects from barriers, the difference becomes more pronounced. The quantum component waves arrive at and reflect from the barriers at different times and the region of self-interference expands. Compounding reflections result in a delocalized jumble of waves. However, remarkably for a matter particle in a square well or subject to a harmonic oscillator potential, partial and complete revival of the original wave packet appears "out of nowhere" periodically due to the rational relationship between the energies of the eigenstates in these systems.

This educational Live Script illustrates how to solve the time-dependent Schrödinger equation in one dimension by applying the method of lines and by an expansion in eigenstates. It illustrates the revival phenomenon for various wave packet shapes describing a particle in a box and subject to a harmonic oscillator potential. The script may interest students and instructors in physics and related fields. 'Try this' suggestions, coding 'Challenges,' hyperlinks, and references are included for further exploration. Additional educational Live Scripts by the author may be found here and some concerning numerical quantum mechanics are listed in the references.