# Unable to decrypt emission/absorption matrix element

Unable to "decrypt" emission/absorption matrix element

Hi to everyone! I'm reading an example of atom radiation interaction (more precisely, an atom emits a photon which photoionizes another atom) on an old book (60s) and I'm facing a matrix element which I never saw. I mean, I'm not able to recognize in it those usual elements you can find in ordinary treatments of photon-atoms transitions. Since the author does not provide any reference from where he derives this kind of notation, I'm not able to understand how he calculates the amplitude probability of photon emission followed by the photon absorption.

If somebody can tell me how to interpret this matrix element or, better, suggest me some reference where I can find the same kind of notation, I'll be very happy and grateful!

For example, the emission matrix element for the atom which emits the photon is:
$$V = C \hat{p} \cdot \hat{A} k^{-\frac{1}{2}} \exp[-i\textbf{k}\cdot\textbf{r}]$$

where:

$$C$$ includes costants which are not important;
$$\hat{p}$$ unit polarization vector of the dipole radiative transition in the atom;
$$\hat{A}$$ polarization vector of the photon
$$\textbf{k}$$ wave vector of the photon
$$\textbf{r}$$ position of the atom

The absorption matrix element is similar: position vector (r') and polarization-dipole-vector (p') referred to the other atom, same wave vector (k) for the photon but a plus sign on the exponential part.
What puzzles me is: where are the initial and final states of the sistem? from where comes $$k^{-\frac{1}{2}}$$? and what is $$\hat{p}$$ if the term $$\exp[-i\textbf{k}\cdot\textbf{r}]$$ is given? usually, this last term is expanded in dipole approximation and one gets a dipole operator (i.e. $$\textbf{D}=e\textbf{r}$$)...

thx for the help

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wave vector $$\textbf{k}$$ is in radiant/second
position vector $$\textbf{r}$$ in $$c$$seconds with c speed of light.