Position Wave Function not normalizable? -Wave function in the Double Slit Experiment

Question

I was going through the Feynman Lectures Vol. III on Quantum Mechanics and stumbled upon a formula in Chapter 3-1 (http://www.feynmanlectures.caltech.edu/III_03.html) that seems very strange to me. Formula 3.7 says that "the amplitude to go from $r1$ to $r2$" is

$$\langle r_2|r_1\rangle=\frac{e^{(ipr_{12}/h)}}{r_{12}}.\tag{3.7} $$

This formula is invoked to explain the interference pattern of the double slit experiment and it makes at least some intuitive sense to me, since the probability of finding an electron on the screen must surely go down as the distance from the middle (and therefore $r_{12}$) increases.

I interpret this as the Position Wave function of the particle, it gives me the amplitude of finding the particle at any point $r_2$.
If that is the case then the wave function would not be normalizable, how can this be?

Also, Feynman claims that the particle has a definite energy and since its a free particle that would imply a definite momentum but i cannot see how this function is supposed to be an eigenfuntion of the momentum operator.

On the other hand, by saying that the particle starts at $r_1$, this would mean that this wave function is the result of a position measurement with the outcome $r_1$ but this also cant be right.

Im also curious what the amplitude would be for a later moment in time, my guess would be something like

$$\langle r_2(t)|r_1\rangle=\frac{e^{(ip*(r_{2}-(r_1+v*t)/h)}}{|r_{2}-(r_1+v*t)|}$$

Which cannot possibly be right since this would imply a kind of classical trajectory for the particle.

I tried to calculate the time evolution of this wave function with mathematica by fourier transforming into the momentum space, applying the hamilton operator on the momentum wave funtion and then fourier transforming it back into the position space (I know how to handle good old wave packets this way) but for this one I just got garbage.

Its seems there is some great misunderstanding of the basics of quantum mechanical position wave functions going on for me here and any help is highly appreciated!

Answer 1

Eq (3.7) is not a wave function, so interpreting it as such is the source of confusion. Rather, it is a transition amplitude, the square of which can be interpreted as a probability (and that does fall as 1/r**2, as you expect). The numerator tracks the phase $ \phi = k\cdot r_{12} = p\cdot r_{12} \ /\hbar $, which is the term describing interference.

Regarding your comment on the Uncertainty Principle: when a wave plane wave passes through a thin slit, its horizontal position uncertainty is the width of the slit--this introduces a transverse momentum uncertainty inversely proportional to the slit size: this is diffraction.