Quantum Harmonic Oscillator — Ladder Operators

Eigenstates ψₙ, probability densities, and coherent state time evolution

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E_n = - ℏω ⟨x⟩ = - ⟨x²⟩ = - Δx·Δp = - ℏ/2
The quantum harmonic oscillator H = ℏω(a†a + 1/2) has eigenstates |n⟩ = (a†)ⁿ|0⟩/√n! with energies E_n = ℏω(n + 1/2). Eigenfunctions ψₙ(x) = H_n(x)e^(-x²/2)/(π^(1/4)√(2ⁿn!)) involve Hermite polynomials. Coherent states |α⟩ = exp(-|α|²/2)Σ(αⁿ/√n!)|n⟩ are minimum uncertainty wavepackets: Δx·Δp = ℏ/2, with ⟨x⟩ = √2 Re(α·e^(-iωt)). Left: wavefunction ψₙ(x) and V(x) = x²/2. Right: probability |ψ|² (or coherent state animation).