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Finite Temperature matrix

  1. Sep 22, 2014 #1
    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. jcsd
  3. Sep 23, 2014 #2

    ShayanJ

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    Gold Member

    [itex] |\psi_1 \rangle [/itex] and [itex] |\psi_2 \rangle [/itex] are normalized eigenvectors of the Hamiltonian. You have the eigenvectors, you just need to normalize them.
    About the factor of [itex] \frac{1}{2z} [/itex], the density operator [itex] \rho [/itex] should have unit trace. But I don't think there should be a factor [itex] \frac 1 2 [/itex] there!
     
  4. Sep 23, 2014 #3
    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.
     
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