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.

 

[paranormal]

I understand your confusion and can provide some guidance on how to approach this problem. First, let's break down the information we have been given. We know that the atom has a total angular momentum of l=1 and is in an eigenstate of Lx with an eigenvalue of \hbar. We also know that it is passing through a region with a uniform magnetic field B in the z-direction, and the Hamiltonian for this system is given. Finally, we are given the matrix representations of the operators Lx, Ly, and Lz.

To express the initial state of the angular momentum in the basis of the eigenstates of the Hamiltonian, we need to find the eigenstates of the Hamiltonian and then express the initial state in terms of these eigenstates. We know that the Hamiltonian operator H is related to the angular momentum operator Lz, so we can use the matrix representation of Lz to find the eigenstates of H. The eigenstates of H will be the basis states that we are looking for.

To find the eigenstates, we can solve the equation Hu_i = \frac{e}{2m_e}BL_zu_i for u_i. This will give us the eigenstates of the Hamiltonian, which we can then use to express the initial state in terms of. Remember that u_i will be a vector in the form [a b c], where a, b, and c are constants. So, when we solve the equation, we will get three eigenstates: u_1, u_2, and u_3.

Once we have the eigenstates, we can express the initial state v in terms of a linear combination of these eigenstates. This means that we can write v as a linear combination of u_1, u_2, and u_3, with coefficients that we will need to solve for. This will give us the basis states that we are looking for.

In summary, to solve this problem, we need to find the eigenstates of the Hamiltonian using the equation Hu_i = \frac{e}{2m_e}BL_zu_i, and then express the initial state v in terms of a linear combination of these eigenstates. I hope this helps and good luck on your exam!