Quantum Mechanics: Linear and Circular polarization states

[Robben]

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:

${\mathbb S}=\left( \begin{array}{cc} \langle x \mid \mathbb{I} \ \mathbb{\hat J}_z \mathbb{I} \mid x \rangle & \langle x \mid \mathbb{I} \ \mathbb{\hat J}_z \mathbb{I} \mid y \rangle \\ \langle y \mid \mathbb{I} \ \mathbb{\hat J}_z \mathbb{I} \mid x \rangle & \langle y \mid \mathbb{I} \ \mathbb{\hat J}_z \mathbb{I} \mid y \rangle \end{array}\right) = \left( \begin{array}{cc} \langle x \mid \pm z\rangle \langle \pm z | \ \mathbb{\hat J}_z |\pm z\rangle \langle \pm z \mid x \rangle & \langle x \mid \pm z\rangle \langle \pm z | \ \mathbb{\hat J}_z |\pm z\rangle \langle \pm z \mid y \rangle \\ \langle y \mid \pm z\rangle \langle \pm z | \ \mathbb{\hat J}_z |\pm z\rangle \langle \pm z \mid x \rangle & \langle y \mid \pm z\rangle \langle \pm z | \ \mathbb{\hat J}_z |\pm z\rangle \langle \pm z \mid y \rangle \end{array}\right) \ ?$

 

[Goddar]

Hi. You need to find out from your coursework what the action of J_z is on |R> and |L>, that's the whole point of using them instead of |x> and |y>...

 

[Robben]

Hi. You need to find out from your coursework what the action of J_z is on |R> and |L>, that's the whole point of using them instead of |x> and |y>...

I am not understanding. So what I did was wrong?

The question asks to express $|x\rangle$ and $|y\rangle$ in terms of $|R\rangle$ and $|L\rangle$.

 

[Goddar]

That you did. But then you need to plug these relations in your matrix instead of what you did in the last part, which is what your question was about right?

 

[Robben]

So I did do it correctly and all I have to do is plug in the equations. Thank you!

 

[Goddar]

Yes but then you'll still have to evaluate the matrix elements! And for that you'll need to know what <R| J_z|R> gives, for example...
Good luck.