The particle in a box — a particle trapped in a 1D infinite square well — is the simplest quantum-confinement model, yet it captures the essence of why confined quantum systems have discrete, quantized energy levels rather than a continuum. It is exactly the physics behind quantum dots: nanocrystal semiconductors a few nanometres across, where confining an electron in a tiny volume sets its energy levels, and hence the colour of light it absorbs or emits — the working principle behind QLED displays and biomedical fluorescent markers. It is also a useful first approximation for electrons in conjugated molecules and semiconductor quantum wells.
E_n = n²π²ħ²/(2mL²) — the quantized energy levels, indexed by
the integer quantum number n.
ψ_n(x) = √(2/L)·sin(nπx/L) — the normalized eigenfunctions of
the infinite square well.
Ψ(x,t) = (1/√2)[ψ_n(x)e^(−iE_n t) + ψ_m(x)e^(−iE_m t)] — a
two-state superposition whose probability density beats at angular
frequency ω = E_m − E_n.
Real quantum dots — nanocrystals only a few nanometres wide — glow different colours purely because of particle-in-a-box-style confinement: smaller dots confine electrons more tightly, raising their energy gap and shifting emitted light toward blue, while larger dots glow red. This exact effect is mass-produced today in QLED televisions and used in biomedical imaging as tunable fluorescent labels.
The particle in a box is the simplest solvable model in quantum mechanics: a particle confined to a 1D region of length L with infinitely high walls at the boundaries. Because the wavefunction must vanish at the walls, only a discrete set of standing-wave shapes fit inside — the eigenfunctions ψn(x) = √(2/L)·sin(nπx/L) — each tied to a quantized energy En = n²π²ħ²/(2mL²). This single result — that confinement forces energy to come in discrete steps rather than a smooth continuum — is the seed from which most of quantum mechanics grows.
This simulation lets you inspect a single eigenstate, showing both the wavefunction ψn(x) and its probability density |ψn(x)|², or switch to a superposition of two eigenstates n and m and watch the probability density genuinely evolve in time, sloshing back and forth across the box at the quantum beat frequency ω = Em − En. The same confinement physics governs real quantum dots — semiconductor nanocrystals only a few nanometres wide — where shrinking the box raises the energy gap and shifts the emitted colour toward blue, the principle used in QLED displays and biomedical imaging labels.
It is an idealised quantum system: a particle confined to a 1D region of length L by infinitely high potential walls at x = 0 and x = L, with zero potential inside. Solving the Schrödinger equation under these boundary conditions gives a discrete ladder of allowed energies rather than a continuum, making it the clearest illustration of quantum energy quantisation.
The wavefunction must be zero at both walls, since the particle cannot exist where the potential is infinite. Only sine waves that fit a whole number of half-wavelengths between the walls satisfy this condition, and each allowed wavelength corresponds to one specific energy En = n²π²ħ²/(2mL²). Boundary conditions, not the equation itself, are what force the energies to be discrete.
It mixes two eigenstates, n and m, into a single time-dependent state Ψ(x,t) and plots its live probability density |Ψ(x,t)|². Because the two components accumulate phase at different rates (set by their energies), the combined density is not static — it visibly oscillates, or "beats", back and forth inside the box at angular frequency ω = Em − En, a direct demonstration of real quantum time evolution.
It is the working principle behind quantum dots — nanocrystal semiconductors a few nanometres across used in QLED televisions, solar cells and biomedical fluorescent imaging — where confining an electron to a nanoscale volume sets its energy gap, and therefore its emitted colour, purely through box-like confinement. It is also the standard first approximation for electrons in conjugated molecules and in engineered semiconductor quantum wells.