The Cooper pair box Hamiltonian in the matrix form

[maximus123]

Hello,

In my problem I need to

Use Matematica (or any other program) to calculate and plot energy bands
(eigenvalues) of the Cooper pair box with (i)E[itex]_C[/itex] = 70, E[itex]_J[/itex] = 10 and (ii) E[itex]_C[/itex] = 20,
E[itex]_J[/itex] = 20

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

 

[QuasiParticle]

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.