Quantum Wave Functions

Particle in a Box

An infinite square well: V=0 inside [0,L], V=∞ outside. Exact solutions exist with energy levels Eₙ = n²π²ħ²/(2mL²).

ψₙ(x) = √(2/L)·sin(nπx/L)
Eₙ = n²·E₁ (energy quanta!)
|ψ|² = probability density
Superposition: coherent mix of levels

State n₁: 1 State n₂ (superposition): 3 Mix ratio (n₁ : n₂): 0.50 Show levels up to: 8

E₁ (ground)1 E₁

E(n₁)—

E(n₂)—

Nodes in ψₙ₁—

Top: wave functions ψ(x). Middle: probability densities |ψ|². Bottom: energy level diagram.