Angular momentum/Hamiltonian operators, magnetic field, basis states problem?

[jeebs]

Hi,
Here's my problem, probably not that difficult in reality but I don't get how to approach it, and I've got an exam coming up soon...

An atom with total angular momentum l=1 is prepared in an eigenstate of L_x, with an eigenvalue of $$\hbar$$. (L_x is the angular momentum operator for the x-component). It then passes into a region where there is a uniform magnetic field B in the z-direction. The Hamiltonian acting on the angular part of the wavefunction in the magnetic field is:
$$H = \frac{e}{2m_e}BL_z$$ where m_e is the electron mass.

I have to "Express the initial state of the angular momentum in the basis of the eigenstate of this Hamiltonian."

I am also given the matrix representation of the operators L_x L_y and L_z:

[ 0 1 0 ]
[ 1 0 1 ]
[ 0 1 0 ]$$\frac{\hbar}{\sqrt{2}} = L_x$$

[ 0 1 0 ]
[-1 0 1 ]
[ 0 -1 0]$$-i\frac{\hbar}{\sqrt{2}} = L_y$$

[1 0 0 ]
[0 0 0 ]
[0 0 -1]$$\hbar = L_z$$

So, I haven't really made much progress with this. What I've thought so far is to start with v being the initial eigenstate the atom is in before the field comes, so that
$$L_xv = \hbar v$$

and then when the field is on, the atom is going to be in one of the eigenstates u_i of the operator H, where $$Hu_i = \frac{e}{2m_e}BL_zu_i$$

Am I actually being asked to write v in terms of a linear combination of u_i's?
If so, then I am stumped as to how to proceed next, and if not, then I haven't got the slightest clue what I'm supposed to be doing.

Can anyone point me in the right direction?
Much appreciated.

 

[kuruman]

First you diagonalize the matrix representing L_x. The state you want is the eigenvector corresponding to eigenvalue $$+\hbar$$.

 

[jeebs]

First you diagonalize the matrix representing L_x. The state you want is the eigenvector corresponding to eigenvalue $$+\hbar$$.

I wasn't sure what diagonalizing meant so I looked on wikipedia. Apparently, if I have an NxN matrix with N eigenvalues, diagonalizing it means writing a diagonal matrix of those 3 eigenvalues, with zeros everywhere else, right?

well, we are only given one eigenvalue for the L_x matrix, namely $$\hbar$$ - how can I make a 3x3 diagonal matrix out of that?
I'm still really not sure what's meant to be going on here...

 

[kuruman]

I wasn't sure what diagonalizing meant so I looked on wikipedia. Apparently, if I have an NxN matrix with N eigenvalues, diagonalizing it means writing a diagonal matrix of those 3 eigenvalues, with zeros everywhere else, right?

well, we are only given one eigenvalue for the L_x matrix, namely $$\hbar$$ - how can I make a 3x3 diagonal matrix out of that?
I'm still really not sure what's meant to be going on here...

Look at this link

http://www.fen.bilkent.edu.tr/~degt/math220/diag.pdf

It provides an example for a 3x3 matrix just like you have. Try to follow the example and if you get stuck let me know.