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

  1. May 17, 2010 #1
    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 [tex]\hbar[/tex]. (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:
    [tex] H = \frac{e}{2m_e}BL_z [/tex] 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 ][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 ui 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 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. jcsd
  3. May 17, 2010 #2

    kuruman

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    First you diagonalize the matrix representing Lx. The state you want is the eigenvector corresponding to eigenvalue [tex]+\hbar[/tex].
     
  4. May 18, 2010 #3
    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 [tex]\hbar[/tex] - 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...
     
  5. May 18, 2010 #4

    kuruman

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    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.
     
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