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

Name: Particle in a Box Energy Calculator
Availability: InStock
Author: Nham Vu

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