Eigenvalues, eigenvectors, eigenstates and operators

[pigletbear]

Homework Statement

Good evening :-)

I have an exam on Wednesday and am working through some past papers. My uni doesn't give the model answers out, and I have come a bit stuck with one question. I have done part one, but not sure where to go from here, would be great if someone could point me in the right direction:

S2) Show that the state vectors |S_x+> = \frac{1}{\sqrt{}2} times a 2x1 matrix (1,1) and |S_y+> = \frac{1}{\sqrt{}2} times a 2x1 matrix (1,-1) are eigenvectors of S_x = h/2 times a 2x2 matrix (0 1, 1 0) with respective eigenvalues plus and minus h/2...


Part two... Of what operator is the state \frac{1}{\sqrt{}2}(|S_x+> + |S_y+>) and eigenstate, and with what eigenvalue...

Any help would be great and much appreciated

Homework Equations

All in question

The Attempt at a Solution

Part 1: This I can do by using |A - λI| = 0, finding the eigenvalues, then using A.v=λv and setting up simutaneous quations to find the eigenvalues.

Part 2... this is where I need help please :-)

 

[TSny]

Hello.

For part (1) you don't need to go through solving for the eigenvalues and eigenvectors. You just want to verify that |S_x+>, say, is an eigenvector of the given matrix. So, just multiply the matrix times the 2x1 vector representing |S_x+> and verify that you get a constant factor times |S_x+>. The constant factor is your eigenvalue.

For part (2), what 2x1 vector do you get when you add \frac{1}{\sqrt{}2}(|S_x+> + |S_y+>) ? Can you recognize it?

[Aside: The notation|S_y+> for the second given vector is a bit odd. It seems like |S_x-> would be more appropriate.]