Complete Classical Theory of Vibrational Energy Exchange

By using purely classical mechanics, the amount of energy ΔE_vib transferred to an harmonic oscillator molecule in a head‐on molecular collision of velocity v₀ is calculated. In the classical limit for small transition probabilities, the quantum‐mechanical probability per collision of going from the ground state to the first excited state is given by $P0→1=(ΔEvib/ℏω)$⁠, where ω is the frequency of the oscillator, and ΔE_vib is the energy classically transferred to the oscillator in one collision. This transition probability is integrated over a normalized distribution of collision velocities to give an average probability for all collisions 〈P_0→1〉. Finally, 〈P_1→0〉 is calculated from the law of microscopic reversibility. The result turns out to be the same as that obtained by Herzfeld,¹ who used a quantum‐mechanical perturbation treatment and went to the classical limit by assuming that the de Broglie wavelength of the molecule colliding with the oscillator is small compared to the range of the intermolecular forces.