Quantum Harmonic Oscillator

Energy eigenstates, ladder operators, and coherent states

Quantum number n

Energy Eₙ = (n+½)ℏω

0.5 ℏω

Display mode Quantum number n = 0 Coherent state α = 2 Coherent phase φ = 0 ω (animation speed)

ψₙ(x) = Hₙ(x)e^−x²/2/√(2ⁿn!√π)
where Hₙ = Hermite polynomial.

Coherent states |α⟩ are Poisson-weighted superpositions — minimum-uncertainty wave packets that oscillate classically without spreading.

The quantum harmonic oscillator has discrete energy levels Eₙ = (n+½)ℏω. Ladder operators â† and â raise/lower the quantum number: â†|n⟩ = √(n+1)|n+1⟩, â|n⟩ = √n|n-1⟩. Coherent states |α⟩ = e^−|α|²/2 Σ (αⁿ/√n!)|n⟩ are eigenstates of â and represent the most classical quantum states — Gaussian wave packets that oscillate without spreading. They minimize the Heisenberg uncertainty ΔxΔp = ℏ/2.