Result

Particle-in-a-box energies

E_{n} = \frac{n ^{2} π ^{2} ℏ ^{2}}{2 m L ^{2}} (n = 1, 2, 3, \dots)

Statement

The allowed energy levels for a particle confined to a one-dimensional box of length L.

Symbols and units

$E_{n}$	nth allowed energy level	J
$n$	quantum number	—
$L$	box length	m
$ℏ$	reduced Planck constant	1.055×10⁻³⁴ J·s
$m$	particle mass	kg

When it applies

An idealized infinite square well — walls of infinite potential, so ψ = 0 exactly at the boundaries.

Don't use when

Assumes an idealized infinite square well with walls of infinite potential — a finite well allows some wave function leakage beyond the walls and gives different, lower energy levels.

Derivation

The boundary condition ψ = 0 at both walls forces only certain wavelengths to fit — exactly the standing-wave condition on a string fixed at both ends — giving discrete allowed energies:

E_{n} = \frac{n ^{2} π ^{2} ℏ ^{2}}{2 m L ^{2}}

Watch out for

The lowest possible energy (n = 1) is not zero — a confined quantum particle can never be perfectly at rest, a direct consequence of the uncertainty principle. Shrinking the box (smaller L) raises every energy level, since E_n ∝ 1/L², unlike a classical particle which could sit still anywhere for free.

Connects to

Normal modes of a fixed string Time-independent Schrödinger equation