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## Homework Statement

Find the eigenvectors and eigenvalues of exp(iπσ

_{x}/2) where σ

_{x}is the x pauli matrix:

10

01

## Homework Equations

I know that σ

_{x}

^{n}= σ

_{x}for odd n

I also know that σ

_{x}

^{n}is for even n:

01

10

I also know that the exponential of a matrix is defined as Σ(1/n!)x

^{n}where the sum runs from n=0 to infinity

## The Attempt at a Solution

Using knowledge of the matrix exponential, I can say that exp(iπσ

_{x}/2) = Σ(1/n!)(iπσ

_{x}/2)

^{n}. I can then split this into two sums, one for even n and one for odd n. This allows me to take the power off the matrices for easier summing. It's then that I get stuck. I've attached a file showing the two sums because it's easier and clearer to show you this way.[/B]

I'd really appreciate any help that can be given.