Particle in a Box

Why the particle-in-a-box matters

Every undergraduate quantum mechanics class opens here. The system has just enough physics to teach quantization, just little enough math to fit on a single chalkboard. Solve a free Schrödinger equation, apply boundary conditions, count standing waves — and you have a complete bound-state spectrum, eigenfunctions and all. Once you see how ψ(0) = ψ(L) = 0 forces sin(nπx/L), the rest of bound-state QM is variations on this theme.

• Pedagogy. The cleanest demonstration that boundary conditions cause quantization. No magic — just sine waves that have to fit. The same logic works for organ pipes and guitar strings; the only difference is what's vibrating (here, ψ rather than air pressure).
• Quantum dots and nanocrystals. Semiconductor nanoparticles (CdSe, CdTe, InP, perovskites) confine excitons in roughly cubic boxes 2–10 nm wide. The optical band gap shifts with size as 1/L² — measured in absorption spectra and exploited commercially in QLED displays and bio-imaging probes.
• Quantum well lasers. Semiconductor laser diodes use planar GaAs/AlGaAs structures where electrons are confined in one direction (well thickness ~5–10 nm). The lasing wavelength is set by the n=1 to n=2 quantum-well transition energy — engineered by tuning L.
• π-electron systems. Conjugated molecules (polyenes, dyes) approximate as 1D particle-in-a-box for delocalized π electrons. Cyanine dyes' colors correlate with chain length — the textbook example of particle-in-a-box predicting molecular spectra to within a factor of two.
• Nuclear shell model (precursor). The 3D infinite square well is the simplest model of nucleons confined to a nucleus. It predicts magic numbers approximately (2, 8, 20, ...) — refined by adding spin-orbit coupling to recover the experimental magic numbers (2, 8, 20, 28, 50, 82, 126).

Setting up and solving

Inside the box (0 < x < L), V(x) = 0. Outside, V = ∞. The Schrödinger equation inside is the free-particle equation:

−(ℏ²/2m) ψ''(x) = E ψ(x)

→ ψ''(x) = −k² ψ(x), with k = √(2mE)/ℏ

→ ψ(x) = A sin(kx) + B cos(kx)

Outside the box, ψ(x) = 0 (infinite walls). For ψ to be continuous: ψ(0) = 0 and ψ(L) = 0. The first forces B = 0. The second forces sin(kL) = 0 → kL = nπ, n = 1, 2, 3, ... (n=0 gives the trivial ψ ≡ 0 — not a state).

Normalize ∫₀^L |ψ_n|² dx = 1 → A = √(2/L). Putting it together:

ψ_n(x) = √(2/L) sin(nπx/L)

E_n = (ℏ²k²)/(2m) = n²π²ℏ²/(2mL²),  n = 1, 2, 3, ...

E_1 = π²ℏ²/(2mL²)   (ground state)
E_2 = 4 E_1
E_3 = 9 E_1
E_n = n² E_1

Numerical examples

System	L	m	E₁	Notes
Electron in atomic-size box	1 Å (10⁻¹⁰ m)	9.11 × 10⁻³¹ kg	37.6 eV	Atomic-scale ionization energies
Electron in CdSe quantum dot	5 nm	~0.13 m_e	~120 meV	Visible emission tuning
Electron in QW laser	10 nm	0.067 m_e (GaAs)	~56 meV	1.55 μm telecom wavelength
Hydrogen atom in 0.1 mm trap	10⁻⁴ m	1.67 × 10⁻²⁷ kg	2 × 10⁻¹⁸ eV	BEC trap, ultracold
1 g ball in 1 cm box	10⁻² m	10⁻³ kg	5 × 10⁻⁴³ J	Vastly negligible (classical)

The 1/L² scaling is dramatic: shrinking a box by 100× lifts E₁ by 10⁴×. This is why quantum effects appear at nanoscale: chemistry happens because L ~ Å gives eV-scale energies (comparable to room temperature thermal scrambling is 25 meV → atoms barely care).

Worked example — confining an electron

An electron is trapped in a box of width L = 1 Å. Compute the photon emitted in the n=2 → n=1 transition.

E₁ = π²ℏ²/(2m_e L²)
   = π² · (1.055 × 10⁻³⁴ J·s)² / (2 · 9.11 × 10⁻³¹ kg · (10⁻¹⁰ m)²)
   = 6.03 × 10⁻¹⁸ J = 37.6 eV

E₂ = 4 E₁ = 150.4 eV
ΔE = E₂ − E₁ = 112.8 eV  (UV / soft X-ray photon)

λ = hc / ΔE = (1240 eV·nm) / 112.8 eV = 11 nm  (extreme UV)

For a 10 nm box (quantum dot), the same calculation gives E₁ ≈ 3.8 meV, ΔE = 11 meV, λ ≈ 110 μm (far-IR) — but realistic dots have additional band-gap energy from the underlying semiconductor, so visible emission requires combining the dot's particle-in-a-box energies with the bulk band gap.

Common mistakes

• Including n = 0. ψ_0(x) = sin(0) = 0 everywhere is not a state — it's no probability anywhere, which violates normalization. The lowest allowed n is 1.
• Forgetting the π². E_n = ℏ²k²/(2m) with k = nπ/L gives the π² factor. Dropping it gives wrong numbers by an order of magnitude.
• Using infinite-well results for finite barriers. Real quantum dots have finite walls; the wave function leaks into the barrier as an exponential decay. Energies are smaller than infinite-well predictions. For deep wells, the lowest states are well-approximated; higher states require corrections.
• Conflating 1D with 3D. A 3D box L×L×L has E_{n_x,n_y,n_z} = (n_x²+n_y²+n_z²)·E_1^{1D}. The ground state has n_x=n_y=n_z=1, energy 3·E_1^{1D}. Higher-level degeneracies appear: E = 6 has (1,1,2), (1,2,1), (2,1,1).
• Believing the particle has definite position. Inside the box, the particle is in a stationary wave state — probability density |ψ_n|² oscillates with x but is constant in time. The classical picture of bouncing back and forth doesn't apply to energy eigenstates; it applies only to superpositions.
• Forgetting orthogonality of different n. ⟨m|n⟩ = (2/L) ∫₀^L sin(mπx/L) sin(nπx/L) dx = δ_mn. Different eigenstates are orthogonal — a measurement in one eigenstate has zero probability of "leaking" into another.