Excitation by photons smaller than energy interval

[osturk]

Hi people,

Time-dependent perturbation theory allows for transitions to excited states, through a sinusoidal perturbation whose frequency is smaller than the energy difference between the states. (That is, $P_{a \rightarrow b}=\frac{sin^{2}[(\omega_{0}-\omega)t/2]}{(\omega_{0}-\omega)^2}$. Although the transition probability falls rapidly, as incident light frequency falls below the natural frequency, it's still non-zero..)

So in the "rare" event of absorption of a photon with insufficient energy, where does the lacking energy come from? Can you comment on the energy conservation in such an event?

 

[Bill_K]

Messiah discusses this in Chap XVII, and there he points out that the nonconservation of energy is only apparent, since it is the unperturbed energy (the energy difference between the original states) which is not conserved, rather than the total energy of the system including the perturbation.

 

[osturk]

thanks for the answer. I only have the 1st volume of Messiah's book, and a google search on the issue returns nothing, so..

... nonconservation of energy is only apparent, since it is the unperturbed energy (the energy difference between the original states) which is not conserved, rather than the total energy of the system including the perturbation.

But, the total energy of the system seems to be increased already.. initially we have a photon with $\hbar\omega$, and then, a system with energy $\hbar\omega_{0}$, but no photon. surely energy should be conserved, but how is it compensated?