We determine the momentum operator by requiring that,
when we operate with
on
,
we get
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

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.

Jim Branson 2013-04-22