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.


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.


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



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].


[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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