The method isn’t really different, it only looks that way. This is because we have CHOSEN the Schr¨odinger representation and in that picture the wave FUNCTION is defined as a function of space and time (i.e. x and t).

So then it is easy to find (x) and (x2). Finding (p) and (p2) is a little harder, but the method for finding the EXPECTATION value is the same. Note that if we chose a momentum - FREQUENCY representation INSTEAD of a space - time representation then the momentum expectation value would look simple and the position one would look more complicated.

The method isn’t really different, it only looks that way. This is because we have chosen the Schr¨odinger representation and in that picture the wave function is defined as a function of space and time (i.e. x and t).

So then it is easy to find (x) and (x2). Finding (p) and (p2) is a little harder, but the method for finding the expectation value is the same. Note that if we chose a momentum - frequency representation instead of a space - time representation then the momentum expectation value would look simple and the position one would look more complicated.