Quantum Harmonic Oscillator
ψₙ(x) = Hₙ(x)·e^(-x²/2) — Energy eigenstates
Quantum number n
3
Display mode
Wavefunction ψₙ(x)
Probability |ψₙ(x)|²
Both overlaid
Multiple states
Superposition
Show potential well
Show classical turning pts
Superposition state n₂
5
Mix coefficient α
0.5
Eₙ = (3 + ½)ℏω = 3.5ℏω
Hamiltonian:
H = p²/2m + mω²x²/2
Eigenenergies:
Eₙ = (n+½)ℏω
Wavefunctions:
ψₙ(x) = (2ⁿn!√π)^(-1/2) · Hₙ(x) · e^(-x²/2)
where Hₙ(x) are
Hermite polynomials
. Zero-point energy E₀ = ½ℏω — consequence of the uncertainty principle Δx·Δp ≥ ℏ/2.
Nodes = n; classical turning points at x = ±√(2n+1).