What is Hermitian operator, expectation value?

Write $\langle f|g\rangle = \int f^*g\,\dd\tau$ for two functions that vanish at the boundaries of the region. An operator is Hermitian if $\langle f|\hat Ag\rangle = \langle \hat Af|g\rangle$ for all such $f$ and $g$. For a normalised state $\psi$ ($\langle\psi|\psi\rangle = 1$), the expectation value of $\hat A$ is $\langle A\rangle = \langle\psi|\hat A\psi\rangle$: the mean of many measurements made on identically prepared systems.