- #1

When there is a circular polarized electric field, it can be interpreted as the real part of a complex field, for example

$$E(t) = E_0( \hat{x}+i\hat{y}) e^{-i\omega t}$$

Now, for some selection rules it is useful to calculate the matrix elements of the dipole operator in the direction of the electric field. If we use this definition that operator is

$$D_x + iD_y$$

while if we use directly the real electric field we get a different operator, that should be the correct one. Anyway, to derive the selection rules, usually this notation is used.

How can this work?