Particle in a One-Dimensional Box A Complete Simple Guide

A particle in a one-dimensional box is the simplest quantum model: a particle moves freely between two impenetrable walls at x = 0 and x = L, and can never be found outside. Inside the box the potential energy is zero, and at the walls it is infinite, so the wave function must vanish exactly at both ends.

Though highly idealised, this model explains why confined particles have discrete energy levels instead of any energy they like. The same idea underlies quantum dots, semiconductor nanostructures and the colours of conjugated dye molecules. It is usually the first Schrödinger equation problem students solve fully.

Key Concepts

• Infinite square well: V(x) = 0 for 0 < x < L, and V = ∞ outside. L is the box width in metres.
• Boundary conditions: ψ(0) = 0 and ψ(L) = 0, because the particle cannot exist in the infinite-potential region.
• Quantum number n: a positive integer 1, 2, 3, … labelling the allowed states. n = 0 is not allowed.
• Nodes: points inside the box where ψ = 0 and the particle is never found. State n has n − 1 internal nodes.
• Zero-point energy: the lowest energy E₁ is greater than zero; a confined particle can never sit perfectly still.

Formulas

Allowed energies: E_n = n² h² / (8 m L²), where m is particle mass in kg and energies are in joules. It assumes a non-relativistic particle and perfectly rigid walls.

Normalised wave functions: ψ_n(x) = √(2/L) sin(nπx/L) for 0 ≤ x ≤ L. The wavelength fits the box as λ_n = 2L/n.

Worked Example

An electron (m = 9.11 × 10⁻³¹ kg) is confined in a box of width L = 1.0 nm. Find E₁ in electron-volts.

1. E₁ = h² / (8 m L²) = (6.626 × 10⁻³⁴)² / [8 × 9.11 × 10⁻³¹ × (1.0 × 10⁻⁹)²].
2. Numerator = 4.39 × 10⁻⁶⁷. Denominator = 8 × 9.11 × 10⁻³¹ × 1.0 × 10⁻¹⁸ = 7.29 × 10⁻⁴⁸.
3. E₁ = 6.02 × 10⁻²⁰ J. Divide by 1.602 × 10⁻¹⁹ J/eV to get E₁ ≈ 0.38 eV. Then E₂ = 4E₁ ≈ 1.5 eV and E₃ = 9E₁ ≈ 3.4 eV, showing levels spread apart as n grows.

Common Misconceptions

• The particle can rest at zero energy. Wrong. n starts at 1, so the minimum energy is positive. Perfect rest would fix both position and momentum, violating uncertainty.
• Probability is uniform in every state. Wrong. Only the average over the box is uniform; |ψ_n|² has peaks and nodes, so some positions are favoured.
• A bigger box means higher energy. Wrong. E_n varies as 1/L², so a wider box lowers the levels and they approach a continuum.

Real-World Applications

Quantum dots used in QLED displays confine electrons in tiny boxes, and changing the box size changes the emitted colour. The model estimates energy levels of electrons in conjugated polymers and dyes, guides design of semiconductor quantum wells for lasers, and approximates neutrons confined in nuclei in introductory nuclear physics.

To summarise, confining a particle forces standing waves, quantised energies proportional to n², and a non-zero ground state energy.