- #1
bowlbase
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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}##