Hi I have a query about the difference of two aspects of the statistical interpretation of quantum mechanics given in the popular introductory quantum mechanics books "Introduction to Quantum Mechanics" by Griffiths and "Quantum Mechanics Concepts and Applications" by Zettili.
In Griffiths we have: Consider an observable $\hat{Q}$, with eigenfunctions $f_{n}(x)$ and associated eigenvalues $q_n$. Thus since the eigenfunctions are complete, we have $\Psi(x,t) = \sum_{n}c_n f_n(x)$ which is $$\Psi(x,t) = \langle x| \Psi \rangle = \sum_n \langle x | f_n \rangle \langle f_n |\Psi \rangle$$ where $c_n = \langle f_n | \Psi \rangle$ and $|c_n|^2$ is the probability of the measurement of $\hat{Q}$ would yield eigenvalue $q_n$. After a measurement of the observable, Griffiths states that the state vector collapses to one of the eigenfunctions $f_n$.
In Zettili, he seems to deal with $c_n$ and eigenvalues $a_n$ as the same thing. He has that with eigenvalues $a_n$ and eigenfunctions $| \psi_n \rangle$ of observable $\hat{A}$ which acts on state vector $|\psi(t) \rangle$ we have $$|\psi(t) \rangle = \sum_n a_n | \psi_n \rangle.$$
As can be seen Zettili omits the coefficients $c_n$ but rather takes the eigenvalues $a_n$ as the coefficient. And thus Zettili states that the probability of getting eigenvalues $a_n$ is: $$P_n(a_n) = \frac{|a_n|^2}{\langle \psi | \psi \rangle}.$$
Which interpretation is correct (or preferable) and mostly commonly used?
Also, Zettili states that the state vector collapses to $a_n| \psi_n \rangle$ where as Griffiths just states that the state vector collapses to the eigenfunction?