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Operator matrix elements, opposite-parity functions

  1. Nov 1, 2011 #1
    1. The problem statement, all variables and given/known data

    For operator
    H=f(r)γ , where γ=[itex]\bigl(\begin{smallmatrix}
    0 & I\\
    I& 0
    \end{smallmatrix}\bigr)[/itex] , f(r) some even function.

    Show that matrix element of this operator

    [itex]<\psi_{a}|H|\psi_{b}>[/itex]

    is non-vanishing only if ψ_a and ψ_b functions are of the opposite parity
    (for example 2s and 2p1/2 functions)

    3. The attempt at a solution

    I can write H as

    H = I H I = Ʃ H[itex]_{nm}[/itex]|psi_n><psi_m|

    so the nm matrix element of H is then given as

    H[itex]_{nm}[/itex] = <psi_a|H|psi_b> = <psi_a|I H I|psi_b> =<psi_a|ƩH[itex]_{nm}[/itex]|psi_n><psi_m||psi_b>

    But I'm not sure how to proceed from here.

    Maybe I should take a different approach. For example first to prove that operator
    H is an odd parity operator , then <psi_a|H|psi_b> is non-vanishing only if
    psi_a and psi_b are of the opposite parity , right ?
    But H is even parity operator since f(r) is even, that is
    f(-r)γ=f(r)γ , so I'm stuck again.

    Can someone please clarify things form me here, how to evaluate matrix elements,
    or is my conclusion that H is even wrong ?
     
  2. jcsd
  3. Nov 1, 2011 #2
    A simple answer like yes or no, on my conclusion that H is even would be
    appreciated.
    Anyone ?
     
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