Last updated: 2026-06-16

Particle in a Box

The particle in a box, also called the infinite square well, is one of the first exactly solvable problems in quantum mechanics. A single particle is trapped in a one-dimensional region of length $L$ by walls so high it can never escape. Solving the Schrödinger equation under that constraint reveals that the particle's energy is not free to take any value — it is restricted to a discrete ladder of levels.

This calculator takes the quantum number $n$, the particle mass $m$ and the box length $L$, and returns the energy $E_n$ of that level together with the de Broglie wavelength $\lambda_n$ of the corresponding standing wave.

How the model works

Because the particle can never be found outside the box, its wavefunction must drop to zero at both walls. That boundary condition is identical to the one for a string fixed at both ends: only standing waves that fit a whole number of half-wavelengths into the length $L$ survive. Each allowed standing wave carries a definite momentum, and therefore a definite energy. Every other wavelength is forbidden, which is exactly why the energy comes out quantized.

The lowest level, n = 1, is the ground state. It has a non-zero energy — the particle can never be completely at rest — which is the simplest illustration of zero-point energy.

Formula

Because $E_n$ scales as $n^2$, the levels spread apart as $n$ grows: $E_2 = 4E_1$, $E_3 = 9E_1$, and so on. Narrowing the box or lowering the mass pushes every level higher.

Worked example

An electron ($m = 9.109 \times 10^{-31}\ \text{kg}$) is confined to a box of length $L = 1\ \text{nm} = 1 \times 10^{-9}\ \text{m}$ in its ground state, n = 1.

Step 1 — Energy:

Step 2 — Wavelength:

The single half-wavelength spans the full width of the box, with nodes at each wall. Entering these values in the calculator reproduces the result.

Energy levels at a glance

Real-world relevance

Although idealized, the particle in a box gives surprisingly good estimates whenever an electron is confined to a small region. It predicts the colour trends of conjugated-molecule dyes, the tunable absorption of quantum dots and nanocrystals, and the energy spacing of electrons in narrow semiconductor wells. As a teaching tool it introduces quantization, zero-point energy and the central role of boundary conditions.

Limitation: idealized walls

The model assumes infinitely high walls and a perfectly one-dimensional region, so the particle is strictly confined and never tunnels out. Real wells have finite depth, allowing some leakage of the wavefunction and slightly lowering the energies. For first estimates and for understanding the physics of confinement, the infinite square well remains the standard starting point.