What is the motivation behind the Operator Postulate? I'm independently attempting to learn the principles of quantum mechanics. One postulate is that for any observable, there is an associated Hermitian operator $A$. Moreover, it states that the only possible results of a measurement are the eigenvalues of $A$, and the only state vector $|\psi\rangle$ that will provide this with absolute certainty is the corresponding eigenvector. Thus,
$$A|{\psi}\rangle = \lambda|\psi\rangle$$
But my question is: what special significance do the eigenvalues and vectors of $A$ have? I'm totally lost on what they're supposed to mean. Also, does the product $A|\psi\rangle$ correspond to any action in the real world? I think I have a lot of misconceptions. Thanks.
 A: The point is that a physical observable corresponds to a basis of state vectors, which are assigned definite values of that observable. For example, position is a physical observable, and a state $|x \rangle$ whose wavefunction is concentrated at $x$ has a definite value of position, namely $x$. 
Now, a Hermitian operator happens to have a basis of eigenvectors, which are associated with eigenvalues. So it is useful to say an observable is a Hermitian operator, just because such an operator is a "package" for the relevant information, and we use operators elsewhere all the time. So we talk about, e.g. the position operator $\hat{x}$, but it's only useful because $\hat{x} |x\rangle = x |x \rangle$. 
This is confusing to a lot of new learners because there's no direct relation between measurements of physical observables, and applications of the corresponding Hermitian operators. For example, $\hat{x} |\psi \rangle$ does not represent the state $|\psi \rangle$ after a measurement of $\hat{x}$. 
A: Experiments on quantum systems suggest that these systems can be, when undisturbed, in any supperposition of possible states in which they find themselves ufter disturbance. So, if there is a measurement of some physical observable, like position, we find particle at this position. But careful measurements suggest that particle can be in a state which is a superposition of many positions, so to say. So only after the measurement the particle is found to be in a definite position.And even though you repeat the measurement in exactly the same way you wont get the same result. Further more, if after the first measurement you find the particle somewhere, all other measurements of this type will give the same result. That means if we measure something and we get an answer in form of a number, then, if we measure the same thing again, we will get the same answer. New measurement of a same thing wont disturbed the system. So, we have: 1. System must be described like a combination of all possible states because, it seems, it CAN be in all of them at the same time. So you need a kind of a vector to describe its state. But this is not enough. You also need a set of possible results of experiments( a set of numbers), when before QM, you needed just one number to represent the state of a system. Like, system could be in a position X and that is it. Now we see that the system can be in a state assocciated with many positions. So, we need a vector to describe it, and we also need vector spaces. 
2. Observable, which is core description of a state(because if you just write some vector, without the observable, you have nothing) then must be some matrix of numbers and it can be represented through the possible results of experiments. Also, to every possible such result belongs a state, which is called eigenstate. Action of measurement throws the system to some such eigenstate but action of the operator on the state vector does not yield an eigenvector. It just contains information about the state which is given through the formula that you wrote. Now, what is important is that we have transition amplitudes or what you might call inner products of different states and these inner products(or scalar products) give you amplitude for a system in one state to be found in some other state. Now acting with an observable to some state and then doing the inner product gives you expectation value for that observable (or you can call it average value) but you have to square the transition amplitudes id est inner products to get probabilities and then sum over all eigenvectors to get the average. 
