- #1

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- Homework Statement
- Find the eigenvalue (first order) and eigenvector (0 order) for the first and second excited state (degenerate) for a perturbated hamiltonian

- Relevant Equations
- ##H' = kxy##

##H = \frac{p_x^2}{2m} +\frac{p_y^2}{2m} + \frac{1}{2}m \omega^2 x^2 + \frac{1}{2}m \omega^2 y^2 ##

Hi,

I have to find the eigenvalue (first order) and eigenvector (0 order) for the first and second excited state (degenerate) for a perturbated hamiltonian.

However, I don't see how to find the eigenvectors.

To find the eigenvalues for the first excited state I build this matrix

##

\begin{pmatrix}

\langle 01 | H'| \rangle 01 & \langle 10| H'| \rangle 01 \\

\langle 01 | H'| \rangle 10 & \langle 10| H'| \rangle 10

\end{pmatrix}

= \frac{k \hbar}{2 m \omega}

\begin{pmatrix}

0 & 1 \\

1 & 0

\end{pmatrix}

##

Thus, I get the eigenvalue ##\pm \frac{k \hbar}{2 m \omega}##, but now I have no idea how to find the eigenvector for the 0 order.

Any help will me appreciate, thank you.

I have to find the eigenvalue (first order) and eigenvector (0 order) for the first and second excited state (degenerate) for a perturbated hamiltonian.

However, I don't see how to find the eigenvectors.

To find the eigenvalues for the first excited state I build this matrix

##

\begin{pmatrix}

\langle 01 | H'| \rangle 01 & \langle 10| H'| \rangle 01 \\

\langle 01 | H'| \rangle 10 & \langle 10| H'| \rangle 10

\end{pmatrix}

= \frac{k \hbar}{2 m \omega}

\begin{pmatrix}

0 & 1 \\

1 & 0

\end{pmatrix}

##

Thus, I get the eigenvalue ##\pm \frac{k \hbar}{2 m \omega}##, but now I have no idea how to find the eigenvector for the 0 order.

Any help will me appreciate, thank you.