Expansion Postulate Quantum Mechanics How does the expansion postulate allow predictions to be made about measurement outcomes?
I understand the postulate as:
$$
 ψ =\sum_{n} a_n φ_n
$$
with coefficients calculated by:
$$
a_n =\int φ_n^*ψdτ.
$$
I think that:
$$
|a_n|^2
$$
is the probability of the system being in state φ, but I do not think that is the answer to the question.
 A: The postulates of quantum mechanics say the following about measurements. Consider a physical quantity represented by an operator $\hat{O}$, whose eigenvalue equation is $\hat{O}\phi_n=\lambda_n\phi_n$, for eigenvalues $\lambda_n$ and eigenstates $\phi_n$. The outcome of a measurement of this physical quantity is then one of the eigenvalues of the associated operator.
If you are measuring this property for an arbitrary state $\Psi$, then expanding this state in the basis of eigenstates of the operator $\hat{O}$ gives:
$$
\Psi=\sum_n a_n\phi_n,
$$
where the expansion coefficients are as you wrote. Then quantum mechanics says that probability of obtaining $\lambda_n$ when measuring the property associated with the operator $\hat{O}$ is:
$$
P(\lambda_n)=|a_n|^2.
$$
For degenerate eigenvalues, you need to sum over the degenerate subspace to obtain the total probability.
A: Yes, $\vert a_n \vert^2$ is not the probability of the system being in the state $\phi_n$. It is the probability that the system (which is with $100\%$ probability in the state $\psi$) will be found in a state $\phi_n$ upon a measurement of an operator whose eigenstates are $\{\phi_n\}$.
