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L commutation relation, Baym

  1. Nov 25, 2009 #1
    In Baym's Lectures on Quantum Mechanics he derives the following formula

    [n.L,L]=ih L x n

    (Where n is a unit vector)

    I follow everything until this line:

    ih(r x (p x n)) + ih((r x n) x p) = ih (r x p) x n

    I can't seem to get this to work out. What properties is he using here?
     
  2. jcsd
  3. Nov 25, 2009 #2

    George Jones

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    Can you show

    0 = n x (r x p) + r x (p x n) + p x (n x r) ?

    What you want would follow from this.
     
  4. Nov 25, 2009 #3
    I understand, but is this identity valid since r and p do not commute? This identity is constructed using B(AC)-C(AB) which seems to change order of operation...
     
  5. Nov 25, 2009 #4

    George Jones

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    Try using

    [tex]\left( A \times B \right)_i = \epsilon_{ijk} A_j B_k[/tex]

    and

    [tex]\epsilon_{ijk}\epsilon_{iab} = \delta_{ja}\delta_{kb} - \delta_{jb}\delta_{ka}[/tex]

    in the three terms in your expression (repeated indices are summed over). Also, n is a triple of numbers, and so commutes with r and p.
     
  6. Nov 26, 2009 #5

    George Jones

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    I have now done the calculation. The identity can be verified by using
     
  7. Dec 3, 2009 #6
    Yes, using that theorem this works. Thanks so much!
     
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