Modern · Experiment

The Quantum Well

Trap a particle between two walls and quantum mechanics takes over: it can only hold certain discrete energies, and at each one it spreads out as a standing wave rather than sitting at a point.

wavefunction / probability imaginary part selected energy level

Controls

Energy En1.50 eV
Ground state E₁0.38 eV
Gap to En+1— eV
Nodes inside well1
de Broglie λ = 2L/n1.00 nm
Confinement quantizes energy: only whole half-waves fit, so E grows as n².
About this experiment ›

What you are looking at

A single particle (say an electron) is trapped in a one-dimensional box — a region of width L bounded by walls so high the particle cannot escape (an "infinite square well"). On the left is the energy ladder of states the particle is allowed to occupy; on the right is the wavefunction for the level you select, drawn across the well. The green curve is either the probability density |ψ|² (where you would be likely to find the particle) or the wave ψ itself.

Why the energy is quantized

In quantum mechanics a particle is described by a wave. To fit inside the box the wave must vanish at both walls — exactly like a guitar string clamped at both ends. Only whole numbers of half-wavelengths fit:
L = n · λ/2  (n = 1, 2, 3, …)
Because momentum is tied to wavelength (de Broglie, p = h/λ), allowing only certain wavelengths allows only certain energies:
En = n² h² / (8 m L²)
The energies are discrete and grow as n² — the gaps get wider as you climb. The particle cannot have just any energy, and it can never be perfectly at rest: even the lowest state n = 1 has a non-zero zero-point energy, a direct consequence of the uncertainty principle.

Standing waves and nodes

Each state ψn(x) = √(2/L)·sin(nπx/L) is a standing wave with n − 1 nodes — points inside the well where the particle is never found. The square |ψn|² gives the probability of finding the particle at each position, replacing the classical idea of a definite location with a spread-out cloud of likelihood.

Confinement and time

Squeeze the well (smaller L) or lighten the particle and every energy shoots up — tighter confinement costs more energy, which is why electrons in small atoms and quantum dots have large, well-spaced levels. A single energy state is "stationary": its |ψ|² does not move in time. But switch on superposition (mixing n and n+1) and the probability cloud sloshes back and forth, oscillating at a frequency set by the energy gap, (En+1 − En)/h — the only way the particle's position can actually change in time.

Things to try

Step through the quantum numbers and count the bumps and nodes. Narrow the well and watch the energy ladder stretch upward. Switch to "Re ψ" to see the wave oscillate in time (the rotating quantum phase), and turn on superposition to watch the probability cloud slosh from wall to wall. Use slow motion to follow it.