QUANTUM HARMONIC OSCILLATOR — LADDER OPERATORS
QUANTUM STATE
Energy level n:
0
Coherent state |α|:
0.0
Coherent state phase θ:
0°
Show potential V(x)=½x²
Show |ψ|² (probability density)
Show Re[ψ] wavefunction
Animate coherent state
a⁻ lower
a⁺ raise
ENERGY E_n = ℏω(n + ½)
E = 0.5 ℏω
⟨x⟩ EXPECTATION
0.000
σ_x (UNCERTAINTY)
—
Ladder operators:
a⁺ = (x̂ − ip̂)/√2 (raising)
a⁻ = (x̂ + ip̂)/√2 (lowering)
a⁻|n⟩ = √n|n−1⟩
a⁺|n⟩ = √(n+1)|n+1⟩
ψₙ(x) = π^{-¼}(2ⁿn!)^{-½} Hₙ(x)e^{-x²/2}
Coherent states
|α⟩ = e^{-|α|²/2}Σ αⁿ/√n! |n⟩ minimize uncertainty and behave classically — they oscillate in the potential.