Harmonic oscillator (1-D)

A particle in a parabolic well V(x) = ½μω²x². The eigenstates are ψ_n(x) = N_n H_n(√α x) e^−αx²/2 (Hermite polynomials, α = μω/ℏ), with equally-spaced energies E_n = (n+½)ℏω. Lengths are in units of √(ℏ/μω) (so α = 1) and energies in units of ℏω. Choose the quantum number n.

quantum number n

classical

Energy levels in the well — V(x) = ½μω²x², E_n = (n+½)ℏω

Equally-spaced levels (ΔE = ℏω) sit in the parabolic well; the zero-point energy is E₀ = ½ℏω. The selected ψ_n(x) is drawn on its own energy line, extending past the classical turning points (dashed) into the classically forbidden region.

zoom to level

Wavefunction ψ_n(x) and density |ψ_n(x)|²

ψ_n(x) — amplitude (n nodes)

|ψ_n(x)|² — vs the classical probability (dashed)

Harmonic oscillator (1-D)

Energy levels in the well — V(x) = ½μω²x², En = (n+½)ℏω

Wavefunction ψn(x) and density |ψn(x)|²

Energy levels in the well — V(x) = ½μω²x², E_n = (n+½)ℏω

Wavefunction ψ_n(x) and density |ψ_n(x)|²