Interpretation of Psi, fourier transforms of psi and dirac delta function?
One thing that confuses me about QM is that Psi(x,t)^2 is supposed to yield the PDF (probability density function) of the location of the particle. Usually this is represented by the Dirac Delta function. If I have attempt to observe only the position of a particle we have a thin vertical line and the fourier transform is a single frequency sine wave supposedly depicting its momentum.
A couple of questions:
1. The dirac delta function tells me the variance of the location of the object is very small (perhaps zero). Am I correct in assuming this means practically nothing is known of the momentum or any of its values? I presume we can still bound the momentum somewhat because we know its not travelling faster than light. Does this sub light speed requirement also put limits on location information?
2. The momentum (zeroinformation) sine wave of momentum. However the square of this sine wave is also a sine wave so are we saying that the probability distribution of momentum is periodic? (This seems like hardly zero information to me.) Eg: Psi(x,t)^2 = sin(kxwt)^2 = (1  cos(2*(kxwt)))/2 which is not a uniform (horizontal flat line) PDF.
3. If this is the wrong interpretation of momentum psi what is the correct interpretation?
