• Support PF! Buy your school textbooks, materials and every day products Here!

The Cooper pair box Hamiltonian in the matrix form

  • #1
50
0
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
 

Answers and Replies

  • #2
18,056
7,420
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?
 
  • #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.
 

Related Threads for: The Cooper pair box Hamiltonian in the matrix form

  • Last Post
Replies
1
Views
4K
Replies
1
Views
683
Replies
5
Views
6K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
5
Views
10K
Replies
1
Views
6K
Replies
2
Views
1K
Top