Suppose you have a state described by the wave function $\psi(x) = \phi_1(x)+2\phi_2(x)+3\phi_3(x)$ , where the $\phi$s are normalised eigenfunctions of a Hermitian operator $\hat{O}$ with eigenvalues $\lambda_1 = 1, \lambda_2 = 5, \lambda_3 = 9$.
What happens to the wave function if you measure $\hat{O}$ and you obtain the result $\lambda_2=5$?
My answer is that the wave function immediately collapses to $\psi(x) = 5\phi_2$
EDIT: My answer should be $\psi(x) = 10\phi_2$, I think
Is that correct? Or should I state the probability with which the wave function will collapse to that state:
$<\hat{O}> = \sum|c_n|^2\lambda_n$ where $|c_n|^2$ is the probability of the $n^{th}$ state.