A quantum ball in a parabolic bowl. Its energy climbs in perfectly even steps and never quite sits still — the most important solvable system in physics.
Quantum Harmonic OscillatorLive
evenly-spaced energy ladder
Controls
Presets
The quantum harmonic oscillator — a particle in a parabolic well — has energy levels evenly spaced by ħω, starting at a nonzero zero-point energy of ½ħω. Its wavefunctions are Hermite polynomials times a Gaussian. It models molecular vibrations, phonons, and quantum fields, making it the most important solvable system in physics.
★ Sign in to save this setupSave your tuned setup, or drop this simulation into your own site, docs, or course page.
How it works
The quantum harmonic oscillator has energy levels E_n = (n + ½)ħω, evenly spaced and starting at a nonzero zero-point energy of ½ħω. Its wavefunctions are Hermite polynomials multiplied by a Gaussian envelope. It models molecular vibrations, lattice phonons, and the modes of quantum fields, appearing everywhere in modern physics.
✦
Ask the AI about this model
The math, the assumptions, real-world uses, or a code translation — explained for this exact simulation.
Is this quantum harmonic oscillator tool really free?▾
Yes. Quantum Harmonic Oscillator 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.