Solution
= Solution
Multiply the <eigenfunction expansion> by $\psi_m^*$ and integrate over the common domain. <Orthogonality> and <wavefunction normalization> give
$$
\int\psi_m^*(x)f(x)\,dx
=\sum_n c_n\int\psi_m^*(x)\psi_n(x)\,dx
=\sum_n c_n\delta_{mn}=c_m.
$$
Thus $\boxed{c_m=\langle\psi_m,f\rangle}$. For a normalized $f$, the <Born rule> identifies $|c_m|^2$ as the probability of this basis outcome.