PX262 - C9b - schmidt orthogonalization

a way of restoring orthogonality
considering $ϕ_{1}$ and $ϕ_{2}$ , which are not orthogonal, but have the same eigenvalue
defining a new eigenfunction, which is a linear combination of the two: $ϕ_{2}^{'} = S_{12} ϕ_{1} - ϕ_{2}$ where $S_{12} = \int ϕ_{1}^{*} ϕ_{2} d τ$
checking if the new function is orthogonal to $ϕ_{1} :$

\begin{aligned} \int ϕ_{1}^{*} ϕ_{2}^{'} d τ & = S_{12} \int ϕ_{1}^{*} ϕ_{1} d τ - \int ϕ_{1}^{*} ϕ_{2} d τ \\ = S_{12} - S_{12} \\ = 0 \end{aligned}

this shows that $ϕ_{1}$ and $ϕ_{2}^{'}$ are orthogonal
if there is another eigenfunction, $ϕ_{3}$ , with the same eigenvalue:
$ϕ_{3}^{'} = s_{13} ϕ_{1} + s_{23} ϕ_{2}^{'} - ϕ_{3}$
where, $s_{13} = \int ϕ_{1}^{*} ϕ_{3} d τ$ , and $s_{23} = \frac{\int ϕ_{2}^{' *} ϕ_{3} d τ}{\int | ϕ_{2}^{'} |^{2} d τ}$
the factors, $s_{12}$ , $s_{13}$ , and $s_{23}$ are called overlap integrals
if there are further degenerate states, the process can be continued until a full set of orthogonal functions is obtained