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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 [tex]\hbar[/tex]. (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:

[tex] H = \frac{e}{2m_e}BL_z [/tex] 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 ][tex]\frac{\hbar}{\sqrt{2}} = L_x[/tex]

[ 0 1 0 ]

[-1 0 1 ]

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

[1 0 0 ]

[0 0 0 ]

[0 0 -1][tex]\hbar = L_z[/tex]

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

[tex] L_xv = \hbar v [/tex]

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 [tex] Hu_i = \frac{e}{2m_e}BL_zu_i [/tex]

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.

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

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