- #1

Robben

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## Homework Statement

Evaluate the matrix elements ##

{\mathbb S}=\left( \begin{array}{cc} \langle x|\mathbb{\hat J}_z|x\rangle& \langle x|\mathbb{\hat J}_z|y\rangle\\

\langle y|\mathbb{\hat J}_z|x\rangle &\langle y|\mathbb{\hat J}_z|y\rangle\end{array}\right)## by expressing the linear polarization states ##|x\rangle## and ##|y\rangle## in terms of the circular polarization states ##|R\rangle## and ##|L\rangle.##

## 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

I worked out ##|x\rangle## and ##|y\rangle## and got:

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

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

To get ##\mathbb{S},## do I just work out:

##

{\mathbb S}=\left( \begin{array}{cc}

\langle x \mid \mathbb{I} \ \mathbb{\hat J}_z \mathbb{I} \mid x \rangle

& \langle x \mid \mathbb{I} \ \mathbb{\hat J}_z \mathbb{I} \mid y \rangle \\

\langle y \mid \mathbb{I} \ \mathbb{\hat J}_z \mathbb{I} \mid x \rangle & \langle y \mid \mathbb{I} \ \mathbb{\hat J}_z \mathbb{I} \mid y \rangle \end{array}\right) =

\left( \begin{array}{cc}

\langle x \mid \pm z\rangle \langle \pm z | \ \mathbb{\hat J}_z |\pm z\rangle \langle \pm z \mid x \rangle

& \langle x \mid \pm z\rangle \langle \pm z | \ \mathbb{\hat J}_z |\pm z\rangle \langle \pm z \mid y \rangle \\

\langle y \mid \pm z\rangle \langle \pm z | \ \mathbb{\hat J}_z |\pm z\rangle \langle \pm z \mid x \rangle & \langle y \mid \pm z\rangle \langle \pm z | \ \mathbb{\hat J}_z |\pm z\rangle \langle \pm z \mid y \rangle \end{array}\right) \ ?

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