Particle in a Box: First Five Energy Levels

Schrödinger 1D infinite well: ψ_n(x) = √(2/L)·sin(nπx/L)

Solve the time-independent Schrödinger equation −(ℏ²/2m)·ψ'' = E·ψ under the boundary conditions ψ(0) = ψ(L) = 0 and you get normalized eigenfunctions ψ_n(x) = √(2/L)·sin(nπx/L), with energies E_n = n²·h²/(8mL²). The probability density |ψ_n(x)|² shows n antinodes (peaks) and n−1 internal nodes (zeros). Take an electron in L = 1 nm: E_1 ≈ 0.376 eV, E_2 ≈ 1.50 eV, E_3 ≈ 3.38 eV, so the levels go as 1 : 4 : 9. Erwin Schrödinger wrote down the equation in 1926 and won the Nobel in 1933. The infinite well is the simplest bound state that isn't trivial, which is why it works as the entry point to atoms, molecules and solids.

Applications

It shows up in quantum chemistry (atomic and molecular orbitals), in solid-state physics (the tight-binding model, band structure), in quantum field theory (mode quantization inside a cavity) and in quantum computing (1D qubits, quantum wires). And of course in teaching, since it's usually the first problem a student can actually solve from scratch.

FAQ

Why is the probability zero at the walls? Outside the box the potential is V = ∞, so ψ has to vanish at x = 0 and x = L. If it didn't, the energy ∫V·|ψ|² would blow up.

What's the most likely position of the particle? For the ground state (n = 1), |ψ_1|² peaks dead center at x = L/2. At n = 2 the peaks move to L/4 and 3L/4, and there's a node sitting at L/2.

Is this realistic? Not exactly; the infinite well is an idealization. Real systems like quantum dots and atoms have walls of finite height, so the wavefunctions leak past them (tunneling), and only a finite number of bound states exist.