# Answer to Question #306 in Quantum Mechanics for hellA

Question #306

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

Expert's answer

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.

∫Ψ*A^Ψdx = ∫ΨA^*Ψ*dx and A∫Ψ*Ψdx = A*∫ΨΨ*dx, whence A=A*, which is possible only when A is real.

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