We determine the momentum operator by requiring that,
when we operate with
times the same wave function.
This means that for these definite momentum states, multiplying by
is the same as multiplying
by the variable
We find that this is true for the following momentum operator.
that this works by explicit calculation.
If we take our momentum operator and act on a arbitrary state,
it gives us the right
for each term in the integral.
This will allow us to compute expectation values for any variable we can represent by an operator.