# Proving two forms of atom-field interaction perturbation Hamiltonian are equivalent

In the presence of an electromagnetic field in the dipole-approximation (${\boldsymbol A} = {\boldsymbol A}(0,t)$) we have the two forms

$$H_{{\boldsymbol d}\cdot {\boldsymbol E}} = - q {\boldsymbol r}\cdot {\boldsymbol E}(t)\,\,\,\,\, ,\,\,\,\, H_{{\boldsymbol p}\cdot {\boldsymbol A}} = - \frac{q}{m} {\boldsymbol p} \cdot {\boldsymbol A}(t)$$

The question asks to show that the first-order time dependent transition probabilities from some $t \to \pm \infty$ are in fact equal for the two choices of the interaction. For a refresher, I'm working with Schiff's notation of the perturbation theory where we are given

$$a_k^{(1)} = \frac{1}{i \hbar} \int\limits_{-\infty}^{\infty} \langle k | H'(t) | n \rangle e^{i \omega_{kn} t'} dt'$$ (The first part of this question asks to prove $\langle k | {\boldsymbol p} | n \rangle = i m \omega_{kn} \langle k | {\boldsymbol r}| n \rangle$ by using Heisenberg's equation of motion and the atomic Hamiltonian $H = \frac{{\boldsymbol p}^2}{2 m} + V({\boldsymbol r})$ ) which is easily done using commutation relations. The only other information we are given is that ${\boldsymbol A}(t)$ is a smooth function on the boundaries of $t \to \infty$.

I've tried a few things by using the relation from the first part in the form $\langle k | {\boldsymbol p}\cdot {\boldsymbol A} | n \rangle = i m \omega_{kn} \langle k | {\boldsymbol r}\cdot {\boldsymbol A} | n \rangle$ and the fact that ${\boldsymbol E} \approx -\frac{\partial {\boldsymbol A}}{\partial t}$ in our dipole-approximation but things are not reducing to what I want them to. Thanks

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I think you will find a discussion of these matters in Cohen-Tannoudji's books on atoms and photons. –  Urgje Apr 13 at 20:46
Do you happen to know which volume and page? –  John M Apr 13 at 20:54
The book is: C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg,Photons and Atoms, Wiley, New York, 1989. I do not know the page but look for gauges. In fact the two expressions for the interaction are unitarily related. An often made mistake in published work is to realise that states are also different in the two representations. I seem to recall that Cohen-Tannoudji cs. warn for this. –  Urgje Apr 14 at 8:01
Afterwards they argue that the equivalence is only ensured for field frequencies equal to that of the considered atomic transition, and they wave their hands for several paragraphs to explain why this should not bother us. It seems rather unsatisfactory to me. The problem is not that they are wrong. Working on another problem (spontaneous emission) I also find that the equivalence is only valid at resonance. I'm more troubled by the fact that this makes the dipole Hamiltonian $H_{{\boldsymbol d}\cdot {\boldsymbol E}}$ useless for off-resonant processes. –  Vinsanity Sep 9 at 15:18
Unfortunately, being retired, I no longer have access to the book. But in my paper, Phys. Rev. A 69, 013804 (2004) I mention this matter in Section X-D and Appendix D. A relevant reference is that to Y. Aharonov and C.K. Au, Phys. Rev. A20, 1553 (1979). I hope this helps somewhat. I can send you the PDF if you want so. –  Urgje Sep 12 at 10:28