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Homework Help: Commutation relation of angular momentum

  1. Dec 26, 2009 #1

    p.p

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    Just wondering if any1 would by any chance know how to calculate the commutation relations when L = R x P is the angular momentum operator of a system?
    [Li, P.R]=?

    P
     
  2. jcsd
  3. Dec 26, 2009 #2

    MathematicalPhysicist

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    Gold Member

    Is it the dot product?
    You can write it down as:
    [Li,PiRi] (using Einstein summation notation).
    and Li=Rj Pk ejki, where ejik is levi-civita tensor.
    and using the basic relation for [R,P]=1 (units of hbar).
     
  4. Dec 27, 2009 #3
    [tex][L_i,p_ir_i]=p_i[L_i,r_i]+[L_i,p_i]r_i[/tex]

    Inserting:
    [tex]L_i=\epsilon_{ijk}r_j p_k[/tex]
    Gives:

    [tex][L_i,p_ir_i]=\epsilon_{ijk}p_i[r_j p_k,r_i]+\epsilon_{ijk}[r_j p_k,p_i]r_i[/tex]
    [tex]=\epsilon_{ijk}p_i r_j[p_k,r_i]+\epsilon_{ijk}[r_j,p_i]r_i p_k[/tex]
    [tex]=-i\hbar\epsilon_{ijk}p_i r_j\delta_{ik}+i\hbar\epsilon_{ijk}\delta_{ij} r_i p_k[/tex]
    [tex]=0[/tex]
     
    Last edited: Dec 27, 2009
  5. Dec 28, 2009 #4

    dextercioby

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    Homework Helper

    The fact that the commutator of the angular momentum and the scalar product of p and r is zero follows naturally. The definition of a vector operator applies for both p and r individually, thus their scalar product should be a scalar operator, therefore the desired commutator is zero.
     
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