Particle in a Box

The simplest quantum system, and the gateway to all of quantum mechanics. Confine a particle and its energy is forced into discrete, quantized levels.

Particle in a BoxLive

quantized energy levels

Controls

Presets

Quantum number n

Well width L (nm)

Show probability density

Confine a quantum particle to a box and its energy can only take discrete values, E_n = n²h²/8mL². The wavefunctions are standing waves with n humps; squaring them gives the probability of finding the particle at each point. Narrowing the box pushes the energy levels dramatically higher — quantum confinement.

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Data Inspector

Energy Eₙ3.384 eV

Level n3

Nodes2

Governing equation

E_{3} = \frac{3 ^{2} h ^{2}}{8 m L ^{2}} = 3.384 eV, ψ_{3} = \frac{2}{L} sin \frac{3 π x}{L}, L = 1 nm

Reading this result: Level n=3 carries 2 interior nodes, and because energy scales as n² it sits 9× above the ground state of this same well.

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How it works

A particle trapped in a one-dimensional infinite well can only occupy energies E_n = n²h²/8mL². Its wavefunctions are standing waves with n antinodes, and their squares give the probability of finding the particle at each position. Shrinking the box raises the energies sharply — the quantum confinement behind quantum dots and nanomaterials.

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Frequently asked questions

Is this particle in a box quantum tool really free?▾

Yes. Particle in a Box runs entirely in your browser using your device's own compute, so local use is free forever. You only pay Compute Tokens if you scale a job to the cloud.

Do I need to install anything?▾

No. Everything runs client-side in a modern browser — no downloads, no license, no account required to start.

Can I save or share my simulation?▾

Create a free account to save projects, and use a shareable embed or minted DOI to publish a live, interactive version anywhere.

How accurate are the results?▾

The solver uses established numerical methods, but results are for research and educational purposes and should be validated against experiment or professional review before you rely on them.