Particle in a box

The model confines a particle of mass m to a region 0 < x < L where the potential is zero; outside this interval the potential is infinite. The time-independent Schrödinger equation reduces to ψ″ = −(2mE/ħ²)ψ, with boundary conditions ψ(0) = ψ(L) = 0. The normalized eigenfunctions are ψₙ(x) = √(2/L) sin(nπx/L) and the energy eigenvalues are Eₙ = n²π²ħ²/(2mL²) for n = 1, 2, 3, ….¹

The quantum number n labels the stationary states; each state has n−1 nodes inside the box, where the probability density vanishes. The ground state (n = 1) has no nodes and possesses a non-zero energy, the zero-point energy, which is a purely quantum effect not present in classical mechanics.

Quantization emerges directly from the boundary conditions: only standing waves with an integer number of half-wavelengths fit in the box. The particle's momentum is correspondingly quantized, and the uncertainty principle dictates that even the ground state has a non-zero kinetic energy, consistent with Δx Δp ≥ ħ/2.

As n grows large, the probability distribution approaches the classical uniform distribution over position, illustrating Bohr's correspondence principle. This model also serves as a stepping stone for understanding more complex systems such as quantum wells, quantum dots, and conjugated molecules in chemistry.²

The one-dimensional box readily extends to two and three dimensions; in a three-dimensional rectangular box the energies are E = (π²ħ²/2m)(nₓ²/Lₓ² + n_y²/L_y² + n_z²/L_z²), leading to degeneracy when different sets of quantum numbers yield the same energy. The particle in a 3D box is a common textbook model for electrons in a metallic nanoparticle or in a quantum dot.

Variants include the finite potential well, where particles can tunnel through the walls, and the so-called 'particle in a box' with a moving wall (the quantum piston or Fermi acceleration problem). These retain the core idea but introduce more physics such as bounded states and tunneling.

A subtle point is that the momentum operator is not self-adjoint on the interval with zero boundary conditions, so the standard momentum has no eigenfunctions satisfying the boundary conditions; this has led to careful discussions about defining momentum observables in a box.³

Another historical curiosity: the particle-in-a-box problem was not fully developed in the early quantum theory; it became a standard pedagogical tool only after the advent of wave mechanics in the late 1920s. Its simplicity helped physicists and chemists understand quantized spectra before more advanced methods were developed.⁴

Potential well

A region of space where the potential energy is lower than its surroundings, confining a particle.

Zero-point energy

The lowest possible energy of a quantum system, which is greater than the classical minimum due to the uncertainty principle.

Boundary conditions

Constraints on the wavefunction at the edges of the region (here ψ=0 at the walls).

Degeneracy

When two or more distinct quantum states have the same energy.