Why Doesn't the Hermitian Calculation Use Complex Conjugates?

[maverick6664]

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

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

\int (\alpha \Delta \hat{A} - i \Delta \hat{B})^* \psi^* (\alpha \Delta \hat{A} - i \Delta \hat{B}) \psi dx <br /> = \int \psi^* (\alpha \Delta \hat{A} + i \Delta \hat{B}) ( \alpha \Delta \hat{A} - i \Delta \hat{B}) \psi dx

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

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

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

Thanks in advance!

 

[Doc Al]

Since the operators are Hermitian, you know that:

\Delta \hat{A}^* = \Delta \hat{A}

\Delta \hat{B}^* = \Delta \hat{B}

 

[Doc Al]

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

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.