Why is energy quantized? When a quantum particle is trapped, can it have any energy — and what decides which energies are allowed?
▶ Launch the interactive simulationSolve the time-independent Schrödinger equation −½ψ'' + V(x)ψ = Eψ by discretizing the Hamiltonian into a symmetric tridiagonal matrix and finding its eigenvalues with Sturm-sequence bisection (machine precision) and its eigenfunctions by inverse iteration. Compare two traps: the infinite square well and the harmonic oscillator V=½x².
only a DISCRETE ladder of energies survives — the wavefunction has to fit the trap, like a string's harmonics — and the SHAPE of the trap sets the spacing law. The box (∞ square well) gives E_n ∝ n² (levels fanning apart, recovered as E_n/E_1 = 1, 4, 9, 16, 25), because the particle is a standing wave with n half-wavelengths. The harmonic oscillator gives a perfectly EVEN ladder E_n = ω(n+½) — and a non-zero ground state E_0 = ½ω, the zero-point energy: Heisenberg forbids the particle from sitting still at the bottom. The energy levels are measured numerically (not assumed) and match the exact formulas to <0.01%