# Properties of operators

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?

$$\int{\phi\hat{A}\psi}=\int{\hat{A}\phi\psi}$$

$$\int{\psi^*(\hat{A}\hat{B})^\dagger\psi}=\int{(\hat{A}\hat{B})^\dagger}\psi^*\psi$$

• for example when the operator is Hermitian

edit:

• $$\dagger$$ here is the Hermitian conjugate
• $$*$$ is the complex conjugate
• $$A^\dagger \ne A^*$$
• "in any case" - that's ambiguous. Do you mean "in every case" or "ever"? 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.