Consider a self-adjoint ('hermitian') operator Aˆ. The wavefunction of the system is ψ. Use the rules forcalculating with scalar products (Chapter 2 in the printed lecture notes, or your notes from my Lecture# 2) to show that the expectation value hAi = hψ|Aψˆ i is real. Do not assume that ψ is an eigenfunction of Aˆ.