I'm trying to show that given $\psi(x)=c_0\psi_0(x)+c_1\psi_1(x)$, where all functions are normalized and additionally that $\psi_0$ and $\psi_1$ are eigenfunctions of an arbitrary operator, that $|c_0|^2+|c_1|^2 = 1$.
From what I understand, the eigenfunctions of a given operator should all be orthogonal to one another (that is $\int_{-\infty}^{\infty}\psi_i(x)^\star\psi_j(x) = 0$, if $i \neq j$), so it makes sense on some level that a linear combination of them that is still normalized in and of itself (specified in the problem) would have a 'magnitude' of 1, so the length of the vector determined by the coefficients $c_0$ and $c_1$ would be 1 (or is my logic totally faulty?).
I'm not sure how to actually demonstrate this, however.