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## Homework Statement

Determine a ##2\times 2## matrix ##\mathbb{S}## that can be used to transform a column vector representing a photon polarization state using the linear polarization vectors ##|x\rangle## and ##|y\rangle## as a basis to one using the circular polarization vectors ##|R\rangle## and ##|L\rangle## as a basis.

## Homework Equations

##|R\rangle = \frac{1}{\sqrt{2}}\left(|x\rangle+i|y\rangle\right)##

##|L\rangle = \frac{1}{\sqrt{2}}\left(|x\rangle-i|y\rangle\right)##

## The Attempt at a Solution

Not sure exactly what the question is asking.

Is it asking to use ##|x\rangle## and ##|y\rangle## as a basis to find the transformation matrix that transforms ##|x\rangle## and ##|y\rangle## to ##|R\rangle## and ##|L\rangle##?

If that is what it's asking, then would this be correct:

##\mathbb{S}=\left[{\begin{array}{cc} \langle R|x\rangle & \langle L|y\rangle \\ \langle R|x\rangle & \langle L|y\rangle \\\end{array}}\right]?##

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