If we have a list of possible outcomeswith the probability of each possible outcome beingthen the value ofwe would expect to get, averaged over many trial isIf the possible outcomes is not in the form of a list of discrete values but a range, so thatmay take any value between certain limitsthe we evaluate an integral. In this casewhere the range of possiblevalues is
In quantum mechanics we have something similar. Remember that the probability density of a particle is given by
whereis the operator of the observable we want to find the expectation value of.
For example ifthe expectation value of the position is
We use the identity,rearranging to obtainThe integral becomes
We integrate by parts:
This is, as we would expect in the centre of the well, sinceis symmetric about the centre of the well.