# Finite Temperature matrix

1. Sep 22, 2014

### bowlbase

1. The problem statement, all variables and given/known data
Consider the Hamiltonian $H=\begin{bmatrix} 0& \frac{-iw}{2}\\ \frac{iw}{2} & 0 \end{bmatrix}$
Write the finite temperature density of the matrix $\rho(T)$

2. Relevant equations
$\beta=\frac{1}{kT}$

3. 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 $z=e^{-\beta E_1}+e^{-\beta E_2}$

2. Sep 23, 2014

### 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!

3. Sep 23, 2014

### 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.