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

1. May 17, 2010

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 Lx, with an eigenvalue of $$\hbar$$. (Lx 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 me 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 Lx Ly and Lz:

[ 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 ui 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 ui'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.

2. May 17, 2010

kuruman

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

3. May 18, 2010

jeebs

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 Lx 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...

4. May 18, 2010