Particle in a Box
The particle in a box is a model of a particle confined to a finite region of space by infinitely high potential walls, whose exact solution to the Schrödinger equation yields discrete, quantised energy levels rather than a continuous range — the simplest exactly solvable problem in quantum mechanics.
Curious
Imagine a ball bouncing between two perfectly rigid walls in classical physics — it can move at any speed you give it, and therefore have any energy at all, continuously. Confine a quantum particle, like an electron, the same way, and something completely different happens: only a specific, discrete set of energies is allowed. The particle cannot have an energy between these values — it's either in one exact allowed state or another, with nothing in between.
This "quantisation from confinement" is not a special rule added on top of quantum mechanics — it falls directly out of solving the Schrödinger equation for a confined particle, the same way a guitar string, tied down at both ends, can only support specific vibration patterns (its fundamental note and overtones) and nothing in between.
Exploring
For a particle of mass m confined to a one-dimensional box of width L (with V = 0 inside and V = ∞ outside), solving the time-independent Schrödinger equation with the boundary condition that the wavefunction must vanish at the walls gives:
The integer n is the quantum number labelling each allowed state. Notice that energy increases with n², so allowed energy levels become more widely spaced as n increases — and the lowest possible energy (n = 1, the ground state) is never zero, a direct consequence of the uncertainty principle: perfectly confining a particle to zero momentum would require perfectly knowing its position, which quantum mechanics forbids.
Worked example: An electron confined to a box of width L = 0.2 nm (roughly atomic scale) has ground-state energy E₁ = (1)²π²(1.0546×10⁻³⁴)² / (2 × 9.109×10⁻³¹ × (0.2×10⁻⁹)²) ≈ 1.5 × 10⁻¹⁸ J ≈ 9.4 eV — comparable to real atomic binding energies, which is why this simple model gives surprisingly reasonable order-of-magnitude estimates for atomic and molecular energy scales despite its extreme simplicity.
Deep Dive
Full derivation. Inside the box (0 < x < L), V = 0, so the time-independent Schrödinger equation reduces to −(ħ²/2m)(d²ψ/dx²) = Eψ, a simple harmonic-oscillator-form differential equation with general solution ψ(x) = A sin(kx) + B cos(kx), where k = √(2mE)/ħ. Outside the box, the infinite potential forces ψ = 0 (a particle cannot exist where the potential is infinite). Continuity of the wavefunction requires ψ(0) = 0, which forces B = 0, leaving ψ(x) = A sin(kx). The second boundary condition, ψ(L) = 0, requires sin(kL) = 0, which is only satisfied when kL = nπ for positive integer n. Substituting back for k and solving for E gives the quantised energy formula above. Normalising ψ (requiring ∫|ψ|²dx = 1 over the box) fixes A = √(2/L), giving the complete normalised wavefunctions ψₙ(x) = √(2/L)·sin(nπx/L).
Generalisation: The infinite square well is the idealised limit of the more physically realistic finite potential well, where the walls have large but finite height. Real confined systems — electrons in quantum dots, nucleons in an approximate nuclear potential — are better modelled by the finite well, in which the wavefunction does not vanish exactly at the walls but instead decays exponentially into the classically forbidden region, a phenomenon closely related to quantum tunnelling.
The particle-in-a-box result is also historically significant as one of the first problems used to teach the general method of solving the Schrödinger equation with boundary conditions, and it remains the standard first example in every introductory quantum mechanics course for exactly that reason.
Concept Relationship Map
Requires: Schrödinger equation, Born interpretation
Enables: quantum harmonic oscillator, finite potential well [PENDING FULL ENTRY], quantum tunnelling
Commonly confused with: the finite potential well (a more realistic model where the walls have finite height)
Sources
- Griffiths, D.J., Schroeter, D.F. (2018). Introduction to Quantum Mechanics, 3rd ed. Cambridge University Press, §2.2.
- Schrödinger, E. (1926). Annalen der Physik, 384(4), 361–376.