- #1
renec112
- 35
- 4
Hi, I am trying to solve an exam question i failed. It's abput pertubation of hydrogen.
I am given the following information:
The matrix representation of [tex] L_y [/tex] is given by:
[tex] L_y = \frac{i \hbar}{\sqrt{2}} \left[\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 \\ 0 & 0 & -1 & 0 \end{array}\right][/tex]
The hydrogen atom is now being pertubed by:
[tex] H' = \alpha L_y[/tex]
Task: Find the first order energyshift for n = 2
[tex] \{\psi_{200}, \psi_{21-1},\psi_{210},\psi_{211}\}[/tex]
with the pertubationMy attempt
I'll have to use degenerte pertubation theory, which states i can find the new energy shift by:
[tex] E^1_{\pm} = \frac{1}{2} ( W_{aa}+W_{bb} \pm \sqrt{(W_{aa}-W_{aa})^2 + 4 W_{ab}|^2}[/tex]
where
[tex] W_{ab} = <\psi^0_a | H' | \psi^0_b > [/tex]
I started by looking at the part
[tex] H' | \psi^0_b > [/tex]
Wich by simple matrix algebra gives me (let's call it A)
[tex] A = \frac{i \alpha \hbar}{\sqrt{2}} \left[\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & \psi_{210} & 0 \\ 0 & -\psi_{21-1} & 0 & \psi_{211} \\ 0 & 0 & -\psi_{210} & 0 \end{array}\right][/tex]
This is where i fail, i think. Because now i have to do some eigenvalue problem at
[tex] W_{ab} = <\psi^0_a | A = \frac{i \alpha \hbar}{\sqrt{2}} [\psi_{200}, \psi_{21-1},\psi_{210},\psi_{211}] \left[\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & \psi_{210} & 0 \\ 0 & -\psi_{21-1} & 0 & \psi_{211} \\ 0 & 0 & -\psi_{210} & 0 \end{array}\right][/tex]
And I'm not sure how i should do it. If i need to do some calculations first or something.
I would very much appreciate some hints :) !
I am given the following information:
The matrix representation of [tex] L_y [/tex] is given by:
[tex] L_y = \frac{i \hbar}{\sqrt{2}} \left[\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 \\ 0 & 0 & -1 & 0 \end{array}\right][/tex]
The hydrogen atom is now being pertubed by:
[tex] H' = \alpha L_y[/tex]
Task: Find the first order energyshift for n = 2
[tex] \{\psi_{200}, \psi_{21-1},\psi_{210},\psi_{211}\}[/tex]
with the pertubationMy attempt
I'll have to use degenerte pertubation theory, which states i can find the new energy shift by:
[tex] E^1_{\pm} = \frac{1}{2} ( W_{aa}+W_{bb} \pm \sqrt{(W_{aa}-W_{aa})^2 + 4 W_{ab}|^2}[/tex]
where
[tex] W_{ab} = <\psi^0_a | H' | \psi^0_b > [/tex]
I started by looking at the part
[tex] H' | \psi^0_b > [/tex]
Wich by simple matrix algebra gives me (let's call it A)
[tex] A = \frac{i \alpha \hbar}{\sqrt{2}} \left[\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & \psi_{210} & 0 \\ 0 & -\psi_{21-1} & 0 & \psi_{211} \\ 0 & 0 & -\psi_{210} & 0 \end{array}\right][/tex]
This is where i fail, i think. Because now i have to do some eigenvalue problem at
[tex] W_{ab} = <\psi^0_a | A = \frac{i \alpha \hbar}{\sqrt{2}} [\psi_{200}, \psi_{21-1},\psi_{210},\psi_{211}] \left[\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & \psi_{210} & 0 \\ 0 & -\psi_{21-1} & 0 & \psi_{211} \\ 0 & 0 & -\psi_{210} & 0 \end{array}\right][/tex]
And I'm not sure how i should do it. If i need to do some calculations first or something.
I would very much appreciate some hints :) !