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Quantum Sandbox: 1D Stationary States

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You can choose from among four 1-dimensional potentials , each with two adjustable parameters. On the left are plots of the potential and the lowest five energy levels, shown as horizontal lines. The ground state is designated as . On the right is a plot of the wavefunction or the probability density for the selected level highlighted in blue.

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The eigenstates are calculated using NDEigensystem. Occasionally, the ordering of energy levels might switch for different choices of parameters.

You can vary the parameters for a potential to see how the eigenvalues and eigenfunctions are changed. The emphasis is on qualitative behavior, rather than computational accuracy.

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Contributed by: S. M. Blinder (February 2024)
Open content licensed under CC BY-NC-SA

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The program produces the five lowest-energy solutions of the Schrödinger equation (in atomic units):

,

for the following potentials:

square well: for ;

oscillator: ;

vee potential: ;

double well: .

This Demonstration was inspired, following a suggestion by Michael Trott, by [1].

Reference

[1] D. V. Chulhai, "Quantum Particle-in-a-Sandbox: A Video Game That Explores the Time-Dependent Wave Function for Any Arbitrary One-Dimensional Potential," Journal of Chemical Education, 2023. doi:10.1021/acs.jchemed.3c00774.

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