Extracting Information from a Wavefunction

In a given quantum system, the wavefunction ψ is said to contain all of the information knowable about that system; the only trick is getting it out and that is where operators come in. A fundamental postulate of quantum mechanics is that every measurable quantity, whether it is total energy, angular momentum, position, etc, has a corresponding quantum mechanical operator. The position and momentum operators are of particular importance since nearly all other relevant operators can be constructed from them.

Sometimes when an operator operates on a wavefunction we get, as a result, a constant multiplied by that wavefunction; in other words, an eigenvalue equation. When we obtain such an eigenvalue equation, the corresponding eigenvalue represents the only value for that observable. When we do not obtain an eigenvalue equation, our observables will not be singly-valued; in other words, the observable will span a range of values.

Although we hope for eigenvalue equations, we can still calculate values for observables, but we are doomed to talk only about probabilities. For example, when the position operator x doesn't give an eigenvalue, we can refer instead to the expectation value for x (represented by and often referred to as the average value).