The exact transition probabilities of quantum-mechanical oscillators calculated by the phase-space method

The ‘phase-space’ method in quantum theory is used to derive exact expressions for the transition probabilities of a perturbed oscillator. Comparison with the approximate results obtained by perturbation methods shows that the latter must be multiplied by an exponential factor exp (− ∊/ℏω), where ∊ is the non-fluctuating part of the work done by the perturbing forces; as long as ∊ is small, exp (− ∊/ℏω) ˜ 1 and only dipole transitions have an appreciable probability. As the perturbation energy increases, however, this is no longer true, and multipole transitions become progressively more probable, the most probable ones being those for which the change in energy is approximately equal to the work done by the perturbing forces.