- #1

- 50

- 0

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 chargeUse 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

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]

[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