The particle in a box: the simplest quantum problem

One particle. One dimension. Two hard walls a distance L apart. Inside the box, no potential energy. Outside, infinite potential. This is the simplest possible quantum-mechanical system, and it teaches most of what a chemist needs to know about wavefunctions, quantisation, and the way particle-mass and box-size control energy scales.

The particle in a box is exactly solvable, and its solutions predict — to within a factor of two — the UV-vis absorption maximum of any linear conjugated molecule, from butadiene to β-carotene. That is more predictive power than the model has any right to have. This guide derives the eigenstates, walks the physics, and works two calculations that connect the maths to observable spectra.

Prerequisites: The Boltzmann distribution. If populations vs temperature makes sense, the quantum states here supply the ε_i that populate.

The Schrödinger equation, boxed

Inside a 1D box of length L (0 ≤ x ≤ L), the potential V = 0. The time-independent Schrödinger equation is:

$-\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} = E \psi$

with the boundary conditions ψ(0) = 0 and ψ(L) = 0 (the walls are impenetrable, so the wavefunction must vanish there).

The general solution to a second-order ODE with a positive coefficient of the second derivative is a sinusoid:

$\psi(x) = A \sin(kx) + B \cos(kx)$

The boundary condition at x = 0 kills the cosine (cos 0 = 1 ≠ 0), so B = 0. The boundary condition at x = L forces sin(kL) = 0, meaning kL = nπ for integer n ≥ 1 (n = 0 gives ψ ≡ 0, no particle). So:

$k_n = \frac{n\pi}{L}, \quad n = 1, 2, 3, ...$

Substituting into the Schrödinger equation gives the energy eigenvalues:

$E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2} = \frac{n^2 h^2}{8mL^2}$

using ℏ = h/(2π).

Two things fell out at once from a half-page derivation: energies are quantised (only integer n allowed), and they scale as n² (the levels get further apart, not closer, as n grows — the opposite of an atomic Rydberg series).

Normalisation

The wavefunction ψ_n(x) = A sin(nπx/L) must be normalised so that ∫₀ᴸ |ψ|² dx = 1. Integration gives A = √(2/L), so the normalised wavefunction is:

$\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n \pi x}{L}\right)$

Probability density |ψ_n|² has n − 1 nodes inside the box (zero at x = 0 and x = L excluded). The ground state (n = 1) has no interior nodes — density peaks at the centre. The first excited state (n = 2) has one interior node at L/2 — density peaks at L/4 and 3L/4. This “more nodes = higher energy” pattern generalises to every quantum system.

The three big consequences

Three quantitative predictions fall out of the eigenenergies:

1. Zero-point energy. The ground state has E₁ = h²/(8mL²), not zero. The lowest allowed energy is strictly positive. This is a purely quantum effect — the Heisenberg uncertainty principle in disguise: to confine a particle within L, you must give it a momentum uncertainty of at least ℏ/L, and hence a kinetic energy of at least ℏ²/(2mL²) ~ h²/(mL²).

2. Energy spacing scales as 1/L². Small box → widely-spaced energies. Large box → closely-spaced energies. In the classical limit L → ∞, the spacing vanishes and the energy becomes continuous. This is the correspondence principle in action.

3. Energy spacing scales as 1/m. Light particles have widely-spaced levels; heavy particles have closely-spaced levels. An electron in a 1 nm box has level spacing ~1 eV (visible photon scale). A proton in the same box has spacing 2000× smaller. This is why electronic transitions are typically eV-scale and nuclear vibrations are ~0.1 eV.

Worked example — the electron in a nanometre-size box

The free-electron model for conjugated polyenes

Butadiene, hexatriene, octatetraene, and every longer polyene of the form (CH=CH)_n are chains of alternating single-double bonds. Because of the double-bond conjugation, the π electrons are delocalised over the length of the chain — treat them approximately as free particles in a 1D box of length L = (number of atoms − 1) × (average C-C bond length ≈ 140 pm) + a few 100 pm at each end for the terminal orbitals.

For hexatriene (three C=C bonds, 6 conjugated π electrons):

• L ≈ 5 × 140 pm = 700 pm (roughly)
• Six π electrons fill n = 1, 2, 3 (each level holds two by Pauli).
• HOMO → LUMO transition: n = 3 → n = 4.
• Photon energy: E₄ − E₃ = (16 − 9) × h²/(8m_eL²) = 7 × 6.03 × 10⁻²⁰ × (1 nm / 0.7 nm)² ≈ 8.6 × 10⁻¹⁹ J ≈ 5.4 eV.

The measured absorption maximum of hexatriene is at 268 nm (4.6 eV; Silverstein 8e). Predicted 5.4 eV vs measured 4.6 — about 15% high, which is remarkable given that we have replaced the entire electronic structure with a naked box. Even more: as the chain lengthens (butadiene → decapentaene), the model correctly predicts a red shift because L grows and level spacing shrinks. β-Carotene (11 C=C bonds, L ≈ 3 nm) has a HOMO → LUMO transition at 450 nm (2.75 eV) — exactly in the visible, which is why carrots are orange.

Worked example — HOMO-LUMO of butadiene

Nodes, orthogonality, and the completeness of the basis

Two features of the box eigenstates generalise to every quantum system:

Nodes and energy. The n-th eigenstate has n − 1 nodes. Higher-energy states oscillate more, giving larger |dψ/dx|² and hence larger kinetic energy. This node-counting rule works for the harmonic oscillator, the hydrogen atom, and molecular orbitals — always.

Orthogonality. Any two different eigenstates satisfy ∫₀ᴸ ψ_n ψ_m dx = 0 (for n ≠ m). This is a general property of eigenstates of a self-adjoint Hamiltonian.

Completeness. Any well-behaved function on [0, L] with the same boundary conditions can be expanded as a linear combination of the ψ_n. This is Fourier’s theorem — the box eigenstates are the Fourier sine basis on [0, L].

Extensions

The particle in a box scales up in obvious ways:

2D and 3D boxes. Energies factorise: E(n_x, n_y, n_z) = (h²/8m)(n_x²/L_x² + n_y²/L_y² + n_z²/L_z²). A cubic 3D box has degenerate levels whenever integer squared sums are equal; the first degeneracy is 1²+2²+2² = 2²+1²+2² = 2²+2²+1² = 9, threefold degenerate.

Finite well. Replace the infinite walls with V₀ < ∞. Wavefunctions can now penetrate the walls (exponentially), and quantum tunnelling emerges. Only a finite number of bound states exist; if V₀ is small enough, only the ground state binds (the origin of “particle in a shallow well” chemistry).

Free electron on a ring. Impose ψ(0) = ψ(L) instead of ψ(0) = ψ(L) = 0. Now the allowed k values are k = 2πn/L for n = 0, ±1, ±2, … This is the standard model for π electrons in benzene — six π electrons on a ring, filling n = 0, ±1 levels, giving a 4n+2 aromatic count for stability. See aromaticity: Hückel’s rule for the connection.

Common pitfalls

Using n = 0. n = 0 gives ψ ≡ 0, which is not a physical state. The lowest allowed quantum number is 1.

Forgetting the mass in the denominator. Energy scales as 1/m. An electron in a box has ~1000× the level spacing of a proton in the same box, and ~10⁶× the level spacing of a heavy nucleus.

Assuming the box length equals the C-C bond length. For a conjugated polyene, the effective box is longer than the average bond length times (n − 1); the terminal π orbitals extend slightly beyond the last atom, and empirical corrections of ~50-100 pm per end give best agreement with experiment.

Ignoring degeneracies. In a symmetric multi-dimensional box (like a cube), several distinct (n_x, n_y, n_z) triples can share the same energy. Level counts must include these.

Practice

Where this leads

The particle in a box is the ground floor of quantum mechanics:

• The harmonic oscillator — replaces hard walls with a parabolic well; gives vibrational spectra (see reading an NMR spectrum for the IR/NMR spin analog).
• The hydrogen atom — 3D central potential; solves in spherical coordinates; recovers the periodic-table structure.
• Bloch electrons in solids — free electrons in a periodic potential; band structure emerges.
• Quantum-confined systems — quantum dots, nanowires, 2D materials (graphene, MoS₂); all analysed as particles in engineered boxes.

Every one of them starts by writing −(ℏ²/2m)∂²ψ/∂x² + V(x)ψ = Eψ and solving with boundary conditions.