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Density Matrix Question

  1. Oct 15, 2011 #1
    Firstly, I have been able to find almost nothing on this kind of question in text books or online anywhere. Most places (including my lecture notes) give at most the definition of the operator and that's all. One page if you're lucky out of a whole book. I'd kill for some examples, if you could direct me anywhere it'd be great. Here's my attempt anyway...

    1. The problem statement, all variables and given/known data
    (i)Show that the for the charge density operator ρ for a quantum mechanical syatem, we have [tex] \rho^2=\rho [/tex] and that [tex] tr(\rho)=N [/tex] where N is the number of particles.
    (ii)Consider a diatomic molecule, of atoms A and B. It's electronic structure can be given by the 1s orbitals of A and B, |A> and |B> (assumed orthonormal). Calculate the associated density matrix expanded over the basis set |A>, |B> as a function of number of electrons N=1,2,3,4. Use the nearest neighbour approximation with on site energies [tex] \epsilon_A=-4eV , \epsilon_B=4eV [/tex] and hopping parameter [tex] \gamma=-3eV [/tex]
    (iii) By using ρ from part 1 evaluate the total energy for N=1,2,3,4.


    2. Relevant equations
    (i) [tex] \rho=\sum_{j}^{occupied} |\psi_j><\psi_j| [/tex]
    (ii)[tex] \begin{bmatrix} \epsilon_A & \gamma \\ \gamma & \epsilon_B \end{bmatrix}\left[ \begin{array}{c} \psi_A \\ \psi_B \end{array} \right]= E\left[ \begin{array}{c} \psi_A \\ \psi_B \end{array} \right] [/tex]
    (iii)[tex] L=Tr(L\rho) [/tex] for an operator L

    3. The attempt at a solution
    (i) I think the first part just comes from applying the definition. [tex] \rho^2=\sum_{j}^{occupied} \sum_{i}^{occupied} |\psi_j><\psi_j|\psi_i><\psi_i|=\sum_{j}^{occupied} \sum_{i}^{occupied} |\psi_j> \delta^i_j <\psi_i| =\sum_{j}^{occupied} |\psi_j> <\psi_j|=\rho [/tex]
    For the second part I've no idea really. I think that in a hand-wavy way it may be that the diagonal elements, i.e. the trace, are the probabilities of finding an electron on an atom say, and these probabilities must sum to the number of particles...but I don't know how to put that into maths.
    (ii) I don't know how to go about this, mainly the fact that the question brings in the number of electrons is confusing me.

    (iii) Using the formula above, I guess the answer comes from [tex] E=Tr(H\rho) [/tex], but I need the density matrix first...I guess I'll try get the answer to part 2 first then think about this.

    Thanks alot for any help, or pointers towards somewhere that can help me
     
  2. jcsd
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