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Commutator of density operator with kinetic energy operator

  1. Oct 14, 2011 #1
    Hey guys,

    maybe you can help me with the following problem. I have to calculate the commutator relations in position representation:
    a) [V,ρ]
    b) [p,ρ]
    c) [p^2,ρ]

    Note that <q'|ρ|q>=ρ(q',q) is a matrix element of the density operator

    I already solved the first one. You just have to apply the potential operator on a matix element of the density operator.
    [V,ρ]=<q'|Vρ|q>-<q'|ρV|q>=...=(V(q')-V(q))*ρ(q',q)

    The rest however is more tricky as the momentum operator is not diagonal in this domain.
    [p,ρ]=<q'|pρ|q>-<q'|ρp|q>=...?
    I got the hint that I should try an integration over an auxiliary variable which should lead to something like <q'|ρ|q''>~δ(q'-q'') (Delta functions)

    A Fourier transformation is NOT necessary as far as I know.

    The result of c should be something like:
    -(d^2/dq'^2-d^2/dq^2)*ρ(q',q)

    Thanks a lot for your help!!!!
     
  2. jcsd
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