PX262 - C3 - orthonormality

orthonormality is an important property of hermitian operators: $\int ϕ_{m}^{*} ϕ_{n} d τ = δ_{m n} = {\begin{cases} 1 & for m = n \\ 0 & for m \neq n \end{cases}$ where, $δ_{m n}$ is kronecker delta

\begin{matrix} \hat{Q} ϕ_{n} = q_{n} ϕ_{n} \\ \int ϕ_{n}^{*} \hat{Q} ϕ_{m} d τ = q_{m} \int ϕ_{n}^{*} ϕ_{m} d τ \\ \int ϕ_{m} {\hat{Q}}^{*} ϕ_{n}^{*} d τ = q_{n} \int ϕ_{m} ϕ_{n}^{*} d τ \end{matrix}

if $\hat{Q}$ is hermitian, the two integrals on the $L H S$ must be the same, so: $either, q_{m} = q_{n}, or, \int ϕ_{n}^{*} ϕ_{m} d τ = 0 for m \neq n$
this implies that the eigenfunctions are degenerate, ie. they correspond to the same eigenvalue