Finite Temperature Density Matrix Calculation

[bowlbase]

Homework Statement

Consider the Hamiltonian $H=\begin{bmatrix} 0& \frac{-iw}{2}\\ \frac{iw}{2} & 0 \end{bmatrix}$
Write the finite temperature density of the matrix $\rho(T)$

Homework Equations

$\beta=\frac{1}{kT}$

The Attempt at a Solution

The initial part of the problem had me find the eigenvectors and eigenvalues. I got $\lambda=\pm\frac{w}{2}$ and eigenvectors $v_1=(-i, 1)$ and $v_1=(i, 1)$

Not quite sure what to do with it from here. I know that $\rho=e^{-\beta E_1} | \psi_1 \rangle \langle \psi_1 |+e^{-\beta E_2} | \psi_2 \rangle \langle \psi_2 |$

I think I remember that the eigenvalues are suppose to be $E_{1,2}$. But I don't know what $\psi_{1,2}$ are suppose to be.

Further, my notes show that once I have the matrix I should have a fraction that looks something like $\frac{1}{2z}$ where $z$ is the sum $z=e^{-\beta E_1}+e^{-\beta E_2}$

 

[ShayanJ]

$|\psi_1 \rangle$ and $|\psi_2 \rangle$ are normalized eigenvectors of the Hamiltonian. You have the eigenvectors, you just need to normalize them.
About the factor of $\frac{1}{2z}$, the density operator $\rho$ should have unit trace. But I don't think there should be a factor $\frac 1 2$ there!

 

[bowlbase]

I got it figured it out. I just couldn't find it in my textbook. Turns out in was in recommended reading I guess. Thanks for the help though.