Proof That the Expectation Value and Eigenvalue of a Hermitian Operator Are Always Real
Theorem
The expectation value of a Hermitian operator is always real.
Proof
Let $A$ be a Hermitian operator. The expectation value of $A$ is
$$ \braket{A \rangle = \int \psi^{\ast}A\psi dx = \langle \psi | A\psi} $$
To show that it is real, it suffices to show that $\braket{\psi | A\psi}-\braket{\psi | A\psi}^{\ast}=0$.
$$\begin{align*} \braket{\psi | A\psi}^{\ast} &= \braket{A\psi | \psi} \\ &= \int (A\psi)^{\ast}\psi dx \\ &= \int \psi^{\ast}A^{\ast}\psi dx \\ &= \int \psi^{\ast} A \psi dx \\ &= \braket{\psi | A \psi} \end{align*}$$
Therefore
$$ \braket{\psi | A\psi}-\braket{\psi | A\psi}^{\ast}=\braket{\psi | A\psi}-\braket{\psi | A\psi}=0 $$
■