The infinite square well is the paradigmatic quantum system. Energy is quantized: Eₙ = n²π²ℏ²/(2mL²). Explore wavefunctions, probability densities, and superpositions.
Energy quantization: Eₙ = n²·E₁ where E₁ = π²ℏ²/(2mL²) ≈ 0.376 eV·(nm/L)²/m for electrons
Wavefunctions: ψₙ(x) = √(2/L)·sin(nπx/L) — orthonormal, zero at walls
Superposition: ψ = α·ψ_{n1} + √(1−α²)·ψ_{n2} — probability density oscillates at frequency (E_{n2}−E_{n1})/ℏ
Real systems: Quantum dots, semiconductor wells, conjugated molecules (particle-in-a-box explains UV absorption of dyes). The n³ degeneracy in 3D explains atomic shell structure.