Worked example: A HCl molecule vibrates with frequency ν = 8.66×10¹³ Hz. Find the zero-point energy and the energy/ wavelength of the photon emitted in the n = 1 → 0 transition. Step 1: ω = 2πν = 5.44×10¹⁴ rad/s; ℏω = 1.055×10⁻³⁴ × 5.44×10¹⁴ = 5.74×10⁻²⁰ J = 0.358 eV. Step 2: Zero-point energy E₀ = ½ℏω = 0.179 eV — the molecule vibrates even at absolute zero. Step 3: Transition 1→0 emits ΔE = ℏω = 0.358 eV (equally spaced levels). Step 4: λ = hc/ΔE = 1240/0.358 ≈ 3460 nm — mid-infrared, matching observed HCl vibrational spectra. Conclusion: molecular vibrations are quantum oscillators with ℏω-scale quanta — spectroscopy reads the ladder directly.