PX262 - C9a - degeneracy

in the case of orthonormality, it was assumed that all eigenvalues are different, which is not always true
in a degenerate system there can be multiple states with distinct eigenfunctions with the same eigenvalue of a given dynamical variable, ie: $q_{m} = q_{n}$ for $\hat{Q} ϕ_{n} = q_{n} ϕ_{n}$ and $\hat{Q} ϕ_{m} = q_{m} ϕ_{m}$
there are two places where the arguments go wrong:
- orthonormality
- compatibility of operators
in orthonormality, the following equation was obtained:

(q_{m} - q_{n}) \int ϕ_{m}^{*} ϕ_{n} d τ = 0

by the fact that there are different eigenvalues $(q_{m} \neq q_{n})$ , it was concluded:
$\int ϕ_{m}^{*} ϕ_{n} d τ = 0 for m \neq n$
in case of degeneracy, $q_{m} = q_{n}$ for $m \neq n$ , so the integral may be non-zero
if there are several eigenfunctions, $ϕ_{n}$ , with the same eigenvalue, $q$ , it can be used to create another wavefunction as a linear superposition, eg: $ψ = \sum_{n} c_{n} ϕ_{n}$
the outcome of measurement of a quantity can be evaluated as:
$\hat{Q} \sum_{n} c_{n} ϕ_{n} = \sum_{n} c_{n} \hat{Q} ϕ_{n} = q \sum_{n} c_{n} ϕ_{n} = q ψ$
this shows that $ψ$ is an eigenfunction of $\hat{Q}$ , with the same eigenvalue, $q$
to check for orthogonality:
$\int ϕ_{m}^{*} ψ d τ = \int ϕ_{m}^{*} \sum_{n} c_{n} ϕ_{n} d τ = \sum_{n} c_{n} \int ϕ_{m}^{*} ϕ_{n} d τ = \sum_{n} c_{n} δ_{m n} = c_{m}$
thus, none of the eigenfunctions used to construct $ψ$ is orthogonal to $ψ$
while in general, the linear combination of eigenfunctions is not an eigenfunction, if the eigenfunctions are degenerate, it will be