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Hermitian calculation question

  1. Dec 28, 2005 #1
    Reading back in my book, Greiner's "QM :an introduction" I found a formula I don't understand.

    Let [tex]\alpha[/tex] be a real number, [tex]\Delta \hat{A}, \Delta \hat{B}[/tex] be Hermitian operators. Now I have

    [tex]\int (\alpha \Delta \hat{A} - i \Delta \hat{B})^* \psi^* (\alpha \Delta \hat{A} - i \Delta \hat{B}) \psi dx
    = \int \psi^* (\alpha \Delta \hat{A} + i \Delta \hat{B}) ( \alpha \Delta \hat{A} - i \Delta \hat{B}) \psi dx[/tex]

    This leads to an expected result to prove Heisenberg's Uncertainty Relations, but I think the right-hand side of this formula should be

    [tex]\int \psi^* (\alpha \Delta \hat{A}^* + i \Delta \hat{B}^*) (\alpha \Delta \hat{A} - i \Delta \hat{B}) \psi dx[/tex]

    I should be wrong, but I don't know why. Operators [tex]\Delta \hat{A}, \Delta \hat{B}[/tex] can be complex... (or are they always real?) So will anyone tell me how or why it's correct?

    Thanks in advance!
    Last edited: Dec 28, 2005
  2. jcsd
  3. Dec 29, 2005 #2

    Doc Al

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    Staff: Mentor

    Since the operators are Hermitian, you know that:

    [tex]\Delta \hat{A}^* = \Delta \hat{A}[/tex]

    [tex]\Delta \hat{B}^* = \Delta \hat{B}[/tex]
  4. Dec 29, 2005 #3

    Doc Al

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    Staff: Mentor

    Just a note for clarity. A Hermitian operator is one that equals its adjoint:
    [tex]\hat{A} = \hat{A}^{\dagger}[/tex]

    This implies that the eigenvalues of a Hermitian operator are real (and of course that their mean values are also real). But the operator can certainly be complex and still be Hermitian.
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