What is the physical implication or significance of a wave function being an eigenfunction of some operator? I understand that if some wavefunction $\psi(x)$ is a an eigenfunction of some operator, say, momentum $-i \hbar \frac{d}{dx}$, then:
$$\hat{p}\psi(x)=p\psi(x) \equiv \hat{p}\left| \psi \right> = p \left| \psi \right>$$
$$\left< \psi | \hat{p} | \psi \right> = \left< \psi | p | \psi \right> = p \left< \psi | \psi \right> = p$$
Then that means that the eigenvalue $p$ is the expectation value of momentum. Cool.
That also therefore means that any linear combination of wavefunctions which are eigenfunctions of momentum with eigenvalue $p$ also have momentum $p$. Cool.
But what does that actually… mean? Why is this important? What sort of physical signficance can I ascribe to this? I'm not seeing why my textbooks and homeworks seem to place so much emphases on these eigenfunctions if I can't seem to find any reason why they are important.
 A: It means you can use the eigenvalue as a label for the state since (in your example) $\Delta p=0$.  Not only is the average value $p$, but there is no fluctuation in this outcome.
Hence for instance we can label hydrogen states by energy, angular momentum, and projection because the eigenstates have “unique” (in the sense they do not fluctuate) values of these quantities.  We can speak of a state having this energy, with this angular momentum etc because if you mak a measurement of energy on such an eigenstate you get a single answer.
A: The possible values that can result from measuring an observable $\hat{O}$ is the observables spectrum $\sigma(\hat{O})$. The spectrum of an operator is the set of its eigenvalues (this is not completely true, but let's just ignore that here).
The states that describe a physical system are vectors in a Hilbert space H: $|\Psi\rangle \in H$. You can always find a complete set of eigenvectors of any observable. This allows you to write any vector $|\Psi\rangle \in H$ as a linear combination of eigenvectors
$|\Psi\rangle = \sum_n c_n |\Phi_n\rangle$, with $\hat{O}|\Phi_n\rangle = \lambda_n |\Phi_n\rangle$ and $c_n = \langle \Phi_n|\Psi\rangle$.
The $c_n$'s in the above sum have a physical meaning: When measuring $\hat{O}$, you will measure $\lambda_n$ with a probability of $|c_n|^2$ (this requires $\sum_n |c_n^2| = 1$).
So if the state of the system is described by an eigenvector of some observable $\hat{O}$ (e.g. $c_1 = 1$, and $c_n = 0\  \forall n\geq 2$), then you know with absolute certainty that the measurement of $\hat{O}$ will yield $\lambda_1$.
