What is the particle-in-a-box model?

It is a simplified quantum mechanical model of a particle confined in a region with perfectly rigid, impenetrable walls (infinite potential outside the box). Inside the box the potential is zero; outside it is infinite. The Schrodinger equation yields discrete (quantized) energy levels.

What are the units for box length?

The calculator accepts lengths in nanometers (nm), which is convenient for quantum systems like quantum dots and nanostructures. One nanometer = 1e-9 meters.

What particle mass should I use?

For electrons (the most common case) the default mass is the electron rest mass, 9.10938e-31 kg. You can override this for other particles, such as protons (1.6726e-27 kg) or effective masses in semiconductors.

What does the quantum number n mean?

n (or nx, ny, nz) is a positive integer (1, 2, 3, ...) specifying which energy level the particle occupies. n = 1 is the ground state. Higher n values correspond to excited states with higher energy.

n = 0 would imply a flat wavefunction that satisfies both boundary conditions only as a trivial zero solution (no particle). The minimum allowed value is n = 1, which gives the non-zero zero-point energy — a purely quantum effect with no classical analog.

What is zero-point energy?

Zero-point energy is the lowest possible energy a quantum particle can have (n = 1). Unlike classical mechanics, a confined quantum particle always has some kinetic energy even at absolute zero temperature. It arises from the Heisenberg uncertainty principle.

What is the potential energy of a particle in a box?

Inside the box the potential energy is zero, so all of the particle's energy is kinetic. Outside the box the potential is infinite, which forces the wavefunction to zero at the walls. That is why the total energy E reported here equals the kinetic energy of the chosen quantum state.

Particle in a Box Energy Calculator

Calculate quantized energy levels for a particle trapped in a 1D, 2D, or 3D infinite potential well using the Schrodinger equation solution.

Parameters

Box Dimension

Box Length L_x (nm)

Particle Mass (kg)

Quantum Number n

Energy Level Result

Enter parameters and click Calculate

First 8 Energy Levels

Ground state = n=1

Calculate to see the energy level table.

n	E (Joules)	E (eV)

Quick Reference

1D Box

E = n²h²/(8mL²)

2D Box

E = h²/(8m)(nx²/Lx² + ny²/Ly²)

3D Box

E = h²/(8m)(nx²/Lx² + ny²/Ly² + nz²/Lz²)

h = 6.62607e-34 J·s (Planck's constant)

m_e = 9.10938e-31 kg (electron mass)

1 eV = 1.60218e-19 J

n = 1, 2, 3, ... (positive integers only)

Summary

The particle-in-a-box (infinite potential well) model is one of the simplest exactly solvable problems in quantum mechanics. It describes a particle of mass m confined within a box of fixed dimensions with infinitely high walls, meaning the particle cannot escape. The allowed energy levels are quantized and depend on the box dimensions, the quantum numbers (n), and the particle mass.

How it works

Select the box dimension: 1D, 2D, or 3D.
Enter the box length (and width/height for 2D/3D).
Enter the particle mass (defaults to an electron).
Enter the quantum numbers n (and nx, ny, nz for 2D/3D).
Click Calculate to see the energy level in both Joules and electron-volts.
Use the energy table to compare multiple quantum states at once.

Use cases

Teaching introductory quantum mechanics energy quantization.
Computing energy levels for electrons confined in nanoscale quantum dots.
Modeling electrons in quantum wells used in semiconductor lasers.
Estimating zero-point energy of a particle in a nanotube.
Verifying textbook particle-in-a-box problem solutions.
Exploring how energy scales with box size and quantum number.
Understanding quantum confinement effects in nanotechnology.
Studying degeneracy of 2D and 3D quantum states.

Frequently Asked Questions

Last updated: 2026-05-23 · Reviewed by Nham Vu