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-whenever a linearly polarized photon at angle [tex]\vartheta[/tex] enters apparatus, it displays "2"

-whenever a linearly polarized photon at angle [tex]\frac{pi}{2}+\vartheta[/tex] enters apparatus, it displays "3"

-for all other polarizations other than above, it displays 2 or 3 with random probabilities.

1. Find the eigenvalues and eigenstates.

2.Find the matrices of of operator A in its eigenbasis and |H> |v> basis.

So for part 1, I believe the eigenvalues are 2 and 3. Then the eigenstates are |[tex]\vartheta[/tex]><[tex]\vartheta[/tex]| and |[tex]\frac{pi}{2}+\vartheta[/tex]><[tex]\frac{pi}{2}+\vartheta[/tex]|.

However, I am not sure how to find the eigenvalues and eigenstates for the third property of the apparatus. I probably has to do with the same eigenvalues but the states I am not sure of.

As for part 2, I do not understand it.

So any help would be appreciated!

Thank You

DoubleMint