I am having trouble understanding properties of operators

For an operator $\hat{A}$ which of the following would be correct:

  1. $\int{(\hat{A}\phi)^*\psi}=\int{\hat{A}^*\phi^*\psi}$
  2. $\int{(\hat{A}\phi)^*\psi}=\int{\phi^*\hat{A}^*\psi}$

Is the following allowed in any case?



  • for example when the operator is Hermitian


  • $\dagger$ here is the Hermitian conjugate
  • $*$ is the complex conjugate
  • $A^\dagger \ne A^*$
  • $\begingroup$ "in any case" - that's ambiguous. Do you mean "in every case" or "ever"? $\endgroup$
    – Bill N
    Jan 20 at 14:00

It's worth knowing that in a vector space over the complex numbers the notion of complex conjugtion is basis dependent. In the diagonal-$\hat x$ basis $|x\rangle$ the momentum operator $\hat p \to -i\partial_x$ would change sign when conjugated, but in the diagonal $\hat p$ basis $| p\rangle$ the momentum operator $\hat p$ is just multiplication by the real number $p$. As a consequence the "complex conjugate" of an operator is not really a definable concept.


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