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I am interested how is polarized light absorbed by a molecule or an atom. Unfortunately, I come to a problem in the derivation where a complex vector in a real space appears. This is something I never seen before and I do not know how to interpret it. Therefore I would like to ask you for help about this issue.

From the harmonic perturbation theory and the dipole approximation we obtain the transition rate between two states, [itex] \vert i> [/itex] - initial and [itex] \vert j> [/itex] - final, and this rate is governed by the following matrix element:

[tex] |<j| \frac{e}{m} \vec{e}\cdot \vec{p}|i>|^2 [/tex]

where e is the electron charge, m is the electron mass, [itex] \vec e [/itex] is light polarization vector (it is an unit vector, having information only about direction) and [itex] \vec p [/itex] is the electron momentum operator in the vector form, for more details see Woodgate's book.

Now, if one assumes that [itex] \vec e = (\cos\alpha,\cos\beta,\cos\gamma) [/itex], where [itex] \alpha [/itex], [itex] \beta [/itex] and [itex] \gamma [/itex] are the angles of the [itex] \vec e [/itex] to the [itex] x [/itex], [itex] y [/itex] and [itex] z [/itex] axes of the coordinate frame and [itex] \vec p = (\hat{p}_x, \hat{p}_y, \hat{p}_z) [/itex], we then obtain the following expression:

[tex] <j|\vec{e}\cdot \vec{p}|i> = \frac{i\Delta E}{\hbar}[\cos\alpha <j|ex|i>+\cos\beta <j|ey|i>+\cos\gamma<j|ez|i>][/tex]

which is equal to:

[tex] <j|\vec{e}\cdot \vec{p}|i> = \cos\alpha D_x+\cos\beta D_y+\cos\gamma D_z= \vec{e}\cdot \vec{D}[/tex]

Where [itex]\vec{D}[/itex] is the vector proportional to the dipole moment vector.

Now this is the point where my problems start. Namely, it is obvious that [itex] \vec{D}[/itex] is a complex vector, and I am not sure if the next expression even has a physical meaning:

[tex]\vec{e}\cdot \vec{D} = |D|\cos\delta[/tex]

Where [itex] \delta [/itex] is the "angle" between the polarization angle [itex]\vec{e}[/itex] (exactly known direction in the real space) and the dipole moment complex vector [itex]\vec{D}[/itex].

This is the result that I obtain from the experiment as well, but unfortunately I am not understanding it well enough. I red some mathematical literature on the topic about angles between complex vectors, but I could not understand it very well. Therefore I wish if anyone could help me about understanding what would be the direction of the dipole moment complex vector?

On the other hand, if we for simplicity assume that the light travels along the [itex] z [/itex] direction, and proceed with the complex value of the [itex]\vec{D}[/itex] we have:

[tex] |<j| \frac{e}{m} \vec{e}\cdot \vec{p}|i>|^2 =[/tex]

[tex] =(\cos\phi D_x + \sin\phi D_y)(\cos\phi D^*_x + \sin\phi D^*_y) =[/tex]

[tex] =\cos^2 \phi |D_x|^2+ \sin^2 \phi |D_y|^2 + (D_x D^*_y+D^*_x D_y)\cos\phi \sin\phi =[/tex]

[tex] =\cos^2 \phi |D_x|^2+ \sin^2 \phi |D_y|^2 + (D_x D^*_y+D^*_x D_y)\cos\phi \sin\phi =[/tex]

[tex] =\cos^2 \phi |D_x|^2+ \sin^2 \phi |D_y|^2 + \sin2\phi \cos(\alpha_1-\beta_1) |D_x||D_y|[/tex]

where [itex] D_x=|D_x| e^{i\alpha_1} [/itex], [itex] D_y=|D_y| e^{i\beta_1} [/itex] and [itex] D_x=|D_x| e^{i\gamma_1} [/itex]. By comparing this result to my measurements there is a problem with the [itex] \sin2\phi [/itex] term, but even more explicitly it is in contradiction to the previously obtained result that has only [itex] \cos\delta[/itex] dependence.

I would really appreciate if anyone could help me about this problem or even give any kind of comment.

Best!