Elliptical Polarization (QM)

Pedvi

1. The problem statement, all variables and given/known data
The state of the photons is:
[tex]|\psi> = \frac{1}{\sqrt{1+r^2}}(|\psi_x> + r\exp{(i\alpha)}|\psi_y>)[/tex]
Where the [tex]|\psi_x>[/tex] and [tex]|\psi_y>[/tex] are the linear polarization states in the x and y direction, respectively.
They are elliptically polarized. I have to give the axes a,b of the ellipse, the angle of the major axis and the direction.


2. Relevant equations
I made a change of "axes" to the right and left circular polarization states:
[tex]|\psi_{R/L}> = \frac{1}{\sqrt{2}}(|\psi_x> \pm i|\psi_y>)[/tex]


3. The attempt at a solution
The result of the change is:
[tex]|\psi> = \frac{1}{\sqrt{2(1+r^2)}}(|\psi_R>(1-ir\exp{i\alpha}) + |\psi_L>(1+ir\exp{i\alpha}))[/tex]

I don't really know how to follow, I don't understand if I have to use the Jones matrices or if there's an other way. I think all the necessary information is there.
Could somebody give me some hints?
(It's the first time I write here, sorry if I've made any mistake).