Can the expectation of an operator be imaginary?

[Tspirit]

Assume $\varPsi$ is an arbitrary quantum state, and $\hat{O}$ is an arbitrary quantum operator, can the expectation $$\int\varPsi^{*}\hat{O}\varPsi$$ be imaginary?

 

[hilbert2]

Not if the operator is hermitian. Otherwise it can be. For instance, think about the situation where $\Psi$ is normalized to unity and the operator is just a multiplication with $i$ (imaginary unit).

 

[Tspirit]

Not if the operator is hermitian. Otherwise it can be. For instance, think about the situation where $\Psi$ is normalized to unity and the operator is just a multiplication with $i$ (imaginary unit).

Thank you very much! I have known the deduction.
According to the definition of Hermitian operator denoted by $\hat{O}$ as follows, $$<f|\hat{O}g>=<\hat{O}f|g>,$$ and making $f=g$, we have $$<f|\hat{O}f>=<\hat{O}f|f>=(<f|\hat{O}f>)^{*},$$ meaning $<f|\hat{O}f>$ is real, which resembles the equation $$a=a^{*},$$ meaning $a$ is real.