Quantum Harmonic Oscillator
Wavefunctions ψₙ(x), probability densities |ψₙ|², and superposition dynamics
Quantum Harmonic Oscillator: One of the few exactly solvable quantum systems. Energy levels are equally spaced: Eₙ = ℏω(n + ½), with a zero-point energy ½ℏω even in the ground state — a consequence of the uncertainty principle. Wavefunctions are ψₙ(x) = (2ⁿ n! √π)^{-½} Hₙ(x) e^{-x²/2}, where Hₙ are Hermite polynomials. The superposition ψ = (1/√2)(ψ₀ + ψ₁) forms a coherent-like wave packet that oscillates with the classical frequency — a bridge between quantum and classical dynamics. This model underpins quantum field theory: each field mode is a harmonic oscillator, and particles are excitation quanta.