PX262 - C9c - compatibility of measurements

taking a case where $ϕ_{1}$ and $ϕ_{2}$ are degenerate eigenfunctions of an operator, $\hat{Q}$ , with an eigenvalue, $q$
taking another operator $\hat{R}$ , which commutes with $\hat{Q}$ , ie: $[\hat{Q}, \hat{R}] = 0$

\hat{Q} \hat{R} ϕ_{1} = \hat{R} \hat{Q} ϕ_{1} = q \hat{R} ϕ_{1}

this shows that $\hat{R} ϕ_{1}$ is also an eigenfunction of the operator $\hat{Q}$
the result of $\hat{R} ϕ_{1}$ will also be an wavefunction, which can be written as a linear combination of some eigenfunctions: $\hat{R} ϕ_{1} = a ϕ_{1} + b ϕ_{2}$
similarly: $\hat{R} ϕ_{2} = c ϕ_{1} + d ϕ_{2}$
considering a wavefunction: $ϕ^{'} = A ϕ_{1} + B ϕ_{2}$

\hat{R} ϕ^{'} = A \hat{R} ϕ_{1} + B \hat{R} ϕ_{2} = A a ϕ_{1} + A b ϕ_{2} + B c ϕ_{1} + B d ϕ_{2}

\begin{matrix} (a - r) A + c B = 0 \\ b A + (d - r) B = 0 \end{matrix}

this has solutions only if: $| \begin{matrix} a - r & c \\ b & d - r \end{matrix} | = 0$
this leads to a quadratic equations leading to two possible eigenvalues of $r$ which are generally distinct
therefore, if there is a degenerate system for one operator, there is generally another operator that can distinguish between the degenerate states