For a while, I had difficulty understanding the differences between an expectation value and the state-vector derived from the Schrodinger equation. My understanding is that the Schrodinger equation returns a state-vector $\mid\Psi(t)\rangle$ that can be used to calculate probabilities for events corresponding to possible values, or eigenvalues, of a system. On the other hand, expectation values allow one to determine, with varying accuracy, what the value of an observable is 'most likely' to be, regardless of whether it is a possible state. For example, take a single spin. If the spin is prepared in some direction and rotated in an arbitrary direction, the spin will be either +1 or -1. However, the expectation value will be $\sigma_n = \langle 0 \rangle$. Solving the Schrodinger equation will give a ket $\mid\Psi(t)\rangle$ that can be used to calculate the probabilities that the spin will be +1 or -1.
Both of these are helpful tools used to find what the value of an observable may or may not be. Is there a mathematical or physical relationship between the two? Or are they just two separate ways of obtaining similar but fundamentally different information?