First order pertubation of L_y operator

[renec112]

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 $$L_y$$ is given by:
$$L_y = \frac{i \hbar}{\sqrt{2}} \left[\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 \\ 0 & 0 & -1 & 0 \end{array}\right]$$

The hydrogen atom is now being pertubed by:
$$H' = \alpha L_y$$

Task: Find the first order energyshift for n = 2
$$\{\psi_{200}, \psi_{21-1},\psi_{210},\psi_{211}\}$$
with the pertubationMy attempt
I'll have to use degenerte pertubation theory, which states i can find the new energy shift by:
$$E^1_{\pm} = \frac{1}{2} ( W_{aa}+W_{bb} \pm \sqrt{(W_{aa}-W_{aa})^2 + 4 W_{ab}|^2}$$
where
$$W_{ab} = <\psi^0_a | H' | \psi^0_b >$$

I started by looking at the part
$$H' | \psi^0_b >$$
which by simple matrix algebra gives me (let's call it A)
$$A = \frac{i \alpha \hbar}{\sqrt{2}} \left[\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & \psi_{210} & 0 \\ 0 & -\psi_{21-1} & 0 & \psi_{211} \\ 0 & 0 & -\psi_{210} & 0 \end{array}\right]$$

This is where i fail, i think. Because now i have to do some eigenvalue problem at
$$W_{ab} = <\psi^0_a | A = \frac{i \alpha \hbar}{\sqrt{2}} [\psi_{200}, \psi_{21-1},\psi_{210},\psi_{211}] \left[\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & \psi_{210} & 0 \\ 0 & -\psi_{21-1} & 0 & \psi_{211} \\ 0 & 0 & -\psi_{210} & 0 \end{array}\right]$$
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 :) !

 

[Orodruin]

It seems to me that you are mixing up what is a matrix element and what is a matrix. In the expression for $W_{ab}$, what you have is a matrix element, not a matrix.

The ket state $|\psi^0_a\rangle$ is not a vector containing the $\psi^0_a$, it is a set of different vectors, taking a different value depending on the value of $a$. For example, in your representation, with $a = 200$, that vector would be $(1,0,0,0)^T$.

Furthermore, the expression you have given for the energy shifts of the degenerate perturbation theory is only valid for a two-level degenerate system while you are dealing with a 4-level degenerate system.

The more general approach to degenerate perturbation theory is to look at the restriction of your perturbation to the degenerate subspace, in your case this is a 4-level system, and find its eigenvalues and eigenvectors.Edit: A $\LaTeX$ hint: < and > are relations and are typeset as such. What you are looking for when constructing bras and kets are \langle and \rangle. Compare $|\psi>$ to $|\psi\rangle$.

 

[renec112]

Thank you very much for the reply Orodruin.
So i understand i'll need to construct the 4-4 matrix and find the eigenvalues and eigenvectors. I should construct this matrix by running
$$W_{ab} = \langle \psi^0_a | H' | \psi^0_b \rangle$$

I tried this but the matrix i am getting out is:
$$\left[\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 1\\ 0 & 0 & 0 & 0 \end{array}\right]$$
So i think I'm failing here.

Just to clarify my mistake, this is my approach to construct the matrix is to first look at, say column 2:
$$W_{ab} = \langle \psi^0_a | \alpha L_y | \psi^0_{21-1} \rangle$$
discard all constants for now, and do basic matrix algebra from $$L_y$$

$$\langle \psi^0_a |L_y | \psi^0_{21-1} \rangle = \langle \psi^0_a | \left[\begin{array}{cccc} 0 & 0 & -\psi_{210} & 0 \end{array}\right]$$

Now i have to try all 4 a's. But i only get something from psi 210, because it's the only orthogonal one. So i can see the only thing giving me something in column two is:
$$\langle \psi^0_{210}| \alpha L_y | \psi^0_{21-1} \rangle = -1$$
+ some constants.

What do you think?

 

[Orodruin]

The procedure seems more or less correct. However, the third column is not correct.

 

[renec112]

The procedure seems more or less correct. However, the third column is not correct.

I see - thank you very much for you help :) !