In Chapter 6, you learned that Hermitian operators ($\hat{A}^\dagger = \hat{A}$) have real eigenvalues and orthogonal eigenstates. In Dirac notation, the eigenvalue equation is $\hat{A}|a_n\rangle = a_n|a_n\rangle$, and Hermiticity means $\langle\phi|\hat{A}|\psi\rangle = \langle\psi|\hat{A}|\phi\ra