How do I find eigenstates and eigenvalues from a spin operator?

[johnpaul543]

Homework Statement

I have a spin operator and have to find the eigenstates from it and then calculate the eigenvalues.
I think I managed to get the eigenvalues but am not sure how to get the eigenstates.

Homework Equations

The Attempt at a Solution

I think I managed to get the eigenvalues out to be ±1 by forming the following matrix

 

[vela]

That matrix is wrong. What is $\lambda$? What does $\sin^{\pm i\lambda}$ even mean?

Have you really never found eigenvectors for a matrix? It's typically covered in lower-division linear algebra and differential equations. It seems a bit strange that you'd be in a course asking you to do this problem without having taken the math courses.

 

[johnpaul543]

That matrix is wrong. What is $\lambda$? What does $\sin^{\pm i\lambda}$ even mean?

Have you really never found eigenvectors for a matrix? It's typically covered in lower-division linear algebra and differential equations. It seems a bit strange that you'd be in a course asking you to do this problem without having taken the math courses.

Apologies the $\sin^{\pm i\lambda}$ is incorrect and meant to say $\sin^{\pm i\varphi}$ and yes I have taken eigenvectors before, just not one similar to this.

 

[blue_leaf77]

meant to say $\sin^{\pm i\varphi}$

That doesn't make sense either.