If we have a list of possible outcomes
with the probability of each possible outcome being
then the value of
we would expect to get, averaged over many trial is
If the possible outcomes is not in the form of a list of discrete values but a range, so thatmay take any value between certain limits
the we evaluate an integral. In this case
where the range of possiblevalues is
In quantum mechanics we have something similar. Remember that the probability density of a particle is given by
where
is the operator of the observable we want to find the expectation value of.
For example if
the expectation value of the position is
We use the identity,
rearranging to obtain
The integral becomes
We integrate by parts:
This is, as we would expect in the centre of the well, since
is symmetric about the centre of the well.