- #1
maximus123
- 48
- 0
Hello,
In my problem I need to
states from 0 to 5. Here is the Hamiltonian we are given
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|)
Which in the matrix form looks like
\begin{pmatrix}<br /> \ddots & & & & &\\<br /> & E_C(0-n_g)^2 & -\frac{E_J}{2} & 0 & 0 &\\<br /> &-\frac{E_J}{2} & E_C(1-n_g)^2 & -\frac{E_J}{2} & 0 &\\<br /> &0 & -\frac{E_J}{2} & E_C(2-n_g)^2 & -\frac{E_J}{2} &\\<br /> &0 & 0 & -\frac{E_J}{2} & E_C(3-n_g)^2 &\\<br /> & & & & &\ddots<br /> \end{pmatrix}
Because we are being asked for this matrix from states 0 to 5 I presume this means
\begin{pmatrix}<br /> E_C(0-n_g)^2 & -\frac{E_J}{2} & 0 & 0 & 0 & 0\\<br /> -\frac{E_J}{2} & E_C(1-n_g)^2 & -\frac{E_J}{2} & 0 & 0 & 0\\<br /> 0 & -\frac{E_J}{2} & E_C(2-n_g)^2 & -\frac{E_J}{2} & 0 & 0\\<br /> 0 & 0 & -\frac{E_J}{2} & E_C(3-n_g)^2 & -\frac{E_J}{2} & 0\\<br /> 0 & 0 & 0 & -\frac{E_J}{2} & E_C(4-n_g)^2 & -\frac{E_J}{2} \\<br /> 0 & 0 & 0 & 0 & -\frac{E_J}{2} & E_C(5-n_g)^2<br /> \end{pmatrix}
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
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
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|)
\begin{pmatrix}<br /> \ddots & & & & &\\<br /> & E_C(0-n_g)^2 & -\frac{E_J}{2} & 0 & 0 &\\<br /> &-\frac{E_J}{2} & E_C(1-n_g)^2 & -\frac{E_J}{2} & 0 &\\<br /> &0 & -\frac{E_J}{2} & E_C(2-n_g)^2 & -\frac{E_J}{2} &\\<br /> &0 & 0 & -\frac{E_J}{2} & E_C(3-n_g)^2 &\\<br /> & & & & &\ddots<br /> \end{pmatrix}
Because we are being asked for this matrix from states 0 to 5 I presume this means
\begin{pmatrix}<br /> E_C(0-n_g)^2 & -\frac{E_J}{2} & 0 & 0 & 0 & 0\\<br /> -\frac{E_J}{2} & E_C(1-n_g)^2 & -\frac{E_J}{2} & 0 & 0 & 0\\<br /> 0 & -\frac{E_J}{2} & E_C(2-n_g)^2 & -\frac{E_J}{2} & 0 & 0\\<br /> 0 & 0 & -\frac{E_J}{2} & E_C(3-n_g)^2 & -\frac{E_J}{2} & 0\\<br /> 0 & 0 & 0 & -\frac{E_J}{2} & E_C(4-n_g)^2 & -\frac{E_J}{2} \\<br /> 0 & 0 & 0 & 0 & -\frac{E_J}{2} & E_C(5-n_g)^2<br /> \end{pmatrix}
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