Density matrix for a diagonally polarized photon

Question

I am struggling with the density matrix for diagonally polarized photons. Can I think of diagonally polarized photons as a mixture of vertically and horizontally polarized photons?

Answer 1

Yes, you can write a diagonally polarized state as a linear superposition of horizontally and vertically polarized states ("mixture" isn't the right term though; it's still a pure state). For instance, the $45^{\circ}$ diagonally polarized state $\lvert \nearrow \rangle$ may be expressed as $$\lvert \nearrow \rangle = \frac{1}{\sqrt 2} \lvert \rightarrow \rangle + \frac{1}{\sqrt 2}\lvert \uparrow \rangle.$$

The density matrix $\rho$ of this state can be written as $\lvert\nearrow\rangle\langle \nearrow\rvert$ (cf. this); try calculating the matrix notation by yourself.