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Homework Help: Help with vector operator Del.

  1. Feb 18, 2013 #1
    1. The problem statement, all variables and given/known data
    In the Pauli theory of the electron, one encounters the expresion:

    (p - eA)X(p - eA

    where ψ is a scalar function, and A is the magnetic vector potential related to the magnetic induction B by B = ∇XA. Given that p = -i∇, show that this expression reduces to ieBψ.

    2. Relevant equations

    pXp = 0 and AXA = 0

    3. The attempt at a solution

    I've come to this:

    -e(pXA + AXp

    but I don't even have a clue where to go next since, for all I know,

    pXA + AXp = -(AXp) + AXp = 0


    Someone's got a clue what I should do next? Am I missing something here?
  2. jcsd
  3. Feb 18, 2013 #2


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    xp-px isn't 0. That's not zero for the same reason. The vector potential is a function of x. x doesn't commute with the x component of the momentum operator.
  4. Feb 18, 2013 #3
    Thank you! I got it now. But now I have another question:


    pXA = (-AXp + pXA)/2 or -AXp + pXA

    I need to use this result to end up the problem but it would only work if the sencod result is true. I used -AXp = pXA to develop it but the 1/2 is not letting me move on with the problem in question.
  5. Feb 18, 2013 #4


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    Don't know. I don't know why you think either one is true.
  6. Feb 18, 2013 #5
    pXA = -AXppXA = (pXA + pXA)/2 = (-AXp + pXA)/2
  7. Feb 18, 2013 #6


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    But p x A isn't generally equal to -A x p. It's true for constant vectors because the components of constant vectors are constant and they commute. p is a vector of differential operators and A is a vector of functions of position. They don't commute.
  8. Feb 18, 2013 #7
    Maaaaaaan, you're right. I guess I'll have to find another way to solve this then. Thank you!
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