References: Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Sections 4.1 – 4.2; Exercises 4.2.2 – 4.2.3.
One of the postulates of quantum mechanics is that the momentum operator in position space is given by
The eigenvector of is and has the property that
If we project this onto position space and use 5 we get
Solving this differential equation and normalizing so that we get
For an arbitrary wave function , if we know its position-space form, we can find its momentum-space version as follows:
This has an interesting consequence if the position-space function is real. The probability density for finding a particle in a state with momentum is , which we can write as
In the fourth line, since and are dummy integration variables, both of which are integrated over the same range, we can simply swap them without changing anything. Note that the derivation relies on being real, since if it were complex we would have
That is, for the position that is the argument of the factor appears as the positive term in the exponential, but in 22 the argument of the complex conjugate wave function is , which appears as the negative term in the exponential.
Thus for any real wave function, the probability of the particle having momentum is equal to the probability of it having , so for such wave functions, the mean momentum is always .
As another example, suppose we have a wave function with a mean momentum , so that
If we now multiply by where is a constant momentum, we can calculate the new mean momentum using 5:
The first integral in the fourth line uses the fact that is constant and is normalized so that