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The Cooper pair box Hamiltonian in the matrix form

  1. Mar 12, 2015 #1
    Hello,

    In my problem I need to
    We are advised to create the Cooper pair box Hamiltonian in a matrix form in the charge basis for charge
    states from 0 to 5. Here is the Hamiltonian we are given

    [itex]H=E_C(n-n_g)^2 \left|n\right\rangle\left\langle n\right|-\frac{E_J}{2}(\left|n\right\rangle\left\langle n+1\right|+\left|n+1\right\rangle\left\langle n\right|)[/itex]
    Which in the matrix form looks like
    [itex]\begin{pmatrix}
    \ddots & & & & &\\
    & E_C(0-n_g)^2 & -\frac{E_J}{2} & 0 & 0 &\\
    &-\frac{E_J}{2} & E_C(1-n_g)^2 & -\frac{E_J}{2} & 0 &\\
    &0 & -\frac{E_J}{2} & E_C(2-n_g)^2 & -\frac{E_J}{2} &\\
    &0 & 0 & -\frac{E_J}{2} & E_C(3-n_g)^2 &\\
    & & & & &\ddots
    \end{pmatrix}[/itex]
    Because we are being asked for this matrix from states 0 to 5 I presume this means

    [itex]\begin{pmatrix}
    E_C(0-n_g)^2 & -\frac{E_J}{2} & 0 & 0 & 0 & 0\\
    -\frac{E_J}{2} & E_C(1-n_g)^2 & -\frac{E_J}{2} & 0 & 0 & 0\\
    0 & -\frac{E_J}{2} & E_C(2-n_g)^2 & -\frac{E_J}{2} & 0 & 0\\
    0 & 0 & -\frac{E_J}{2} & E_C(3-n_g)^2 & -\frac{E_J}{2} & 0\\
    0 & 0 & 0 & -\frac{E_J}{2} & E_C(4-n_g)^2 & -\frac{E_J}{2} \\
    0 & 0 & 0 & 0 & -\frac{E_J}{2} & E_C(5-n_g)^2
    \end{pmatrix}[/itex]

    It is then suggested we put this into Mathematica and use the Eigenvalues function to return the eigenvalues so we can then plot the energy bands. I have tried using Mathematica with this matrix but am not getting any results I understand. Is there a method for finding the eigenvalues of this matrix by hand? I am quite lost with this question, any help would be greatly appreciated. Thanks
     
  2. jcsd
  3. Mar 17, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
  4. Mar 18, 2015 #3
    I don't know if this thread is relevant anymore, since it is already a week old. But what was the problem with using the Eigenvalues[] function? It should be quite straightforward. The Mathematica documentation can assist you in using it.
     
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