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Adjunct operator proof (simple!) please tell me if I am right.

  1. Mar 27, 2010 #1
    1. The problem statement, all variables and given/known data

    Hi, I am trying to show that ([tex] \lambda \hat{1})^{t} = \lambda^* \hat{1} [/tex]

    where [tex] \lambda \in [/tex] C (complex numbers)

    and [tex] \hat{1} [/tex] is the identity operator.

    3. The attempt at a solution

    [tex]\int \Psi^* (\lambda \hat{1} )^{t} \Psi d^{3} {r} = \int (\lambda \hat{1} \Psi)^* \Psi d^{3} {r}[/tex]

    = [tex]\lambda^*\int (\hat{1}\Psi)^* \Psi d^{3} {r} [/tex]

    = [tex]\int \Psi^* \lambda^*\hat{1}^{t} \Psi d^{3} {r} [/tex]

    finally, since [tex]\hat{1}[/tex] is hermitian, [tex]\hat{1}^{t}[/tex] = [tex]\hat{1}[/tex]

    so

    [tex]\int \Psi^* (\lambda\hat{1})^{t} \Psi d^{3} {r}[/tex] = [tex]\int \Psi^* \lambda^*\hat{1} \Psi d^{3} {r}[/tex]

    Am I correct? (please excuse my multiple re-edits, I am learning the notation on the fly)
     
    Last edited: Mar 28, 2010
  2. jcsd
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