Physical meaning of quantum operators

Let's say we have a wavefunction $\psi$ and a measurement operator $\hat A$.

I understand how eigenvalues and eigenvectors of $\hat A$ describe the possible outcomes of the measurement.

I also understand that the average measurement can be computed as $\langle \psi|\hat A|\psi\rangle$.

It's still not clear to me what the direct meaning of $\hat A |\psi\rangle$ is. It is a wavefunction; how does its corresponding quantum state relate (in physical terms) to the original state $\psi$ and the measurement $\hat A$?

Answer

I do not think that the action $A\psi$ has a direct physical meaning, when $A$ is a generic observable.

This is because the interpretation of a quantum system as a mathematical model yields the wavefunction and its corresponding Hilbert space as a sort of byproduct. In fact, the state may not always be a wavefunction: without entering too much into details, let's say it is just a mathematical object suitable to evaluate observables.

The mathematical objects with direct physical relevance are observables and states; and the action of an observable on the state (or vice-versa) is assumed to be the evaluation (averaging) process.

Nevertheless, since this (abstract) mathematical system that has QM as a model corresponds exactly to the Hilbert structure of wavefunctions and self-adjoint operators, it may be useful and important to study the behavior of $A\psi$, in order to improve the knowledge of the system, as well as to make physical predictions.

For example, the behavior of $H\psi$, where $H$ is the Hamiltonian operator (energy observable), is directly related with the time evolution of the system (by Schrödinger equation).