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Homework Help: Commutator help

  1. Jun 13, 2007 #1
    1. The problem statement, all variables and given/known data

    I need to show the commutation between the spin operator and a uniform magnetic field will produce the same result as the cross product between them.
    Does this make sense? I don't see how it can be possible.

    2. Relevant equations

    [s,B]

    (The s should also have a hat on it)

    3. The attempt at a solution

    I have sB - Bs but do i represent s as (sx,sy,sz)? x,y,z are subscripts...
    Even if I do that wouldn't the commutation = 0?
     
  2. jcsd
  3. Jun 13, 2007 #2

    George Jones

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    Represent both spin and the magnetic field in terms of Pauli spin matrices.
     
  4. Jun 13, 2007 #3

    nrqed

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    As stated, the question does not quite make sense. I think you mean the commutator of the spin with the hamiltonian of a particle in a uniform B field, [itex] H = \vec{s} \cdot \vec{B} [/itex] . Then you simply have to use the commutation relation of the Pauli matrices [itex] [S_i,S_j] = i \epsilon_{ijk} S_k [/itex] and the result follows trivially (except that it seems to me that one gets "i" times the cross product)

    Patrick
     
  5. Jun 13, 2007 #4
    nrqed:
    So I evaluate [H,s]? In doing that, why would I need the commutation relation [tex] [S_i,S_j] = i \epsilon_{ijk} S_k [/tex] ? It shouldn't be needed if the product terms are only between terms of H and [tex] s_x, s_y, s_z [/tex].
     
  6. Jun 13, 2007 #5

    nrqed

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    But H contains the spin!! See my post.
     
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