Question

Prove that if the operator A^ is Hermitian, its eigenvalues are real.

Answer 1

Let Ψ be an arbitrary eigenfunction of the operator А^ corresponding to its eigenvalue A. Then, due to the self-adjointness of the operator:
∫Ψ*A^Ψdx = ∫ΨA^*Ψ*dx and A∫Ψ*Ψdx = A*∫ΨΨ*dx, whence A=A*, which is possible only when A is real.