Do Quantum Mechanics PDFs Converge?

[LarryS]

Consider the double-slit experiment using a low-intensity source of monochromatic photons: If the intensity is low enough, one can observe the photons, one at a time, slowly developing an interference pattern on a photographic plate.

The interference pattern, over time, resembles more and more the continuous probability density function associated with the wave function, i.e. it “converges” to the PDF.

I have just started reading about the different types of convergence in probability theory:

Convergence in Distribution
Convergence in Probability
Almost Sure Convergence
Etc.

Has anybody found QM literature researching the types of convergence that occur in QM random processes?

Thanks in advance.

 

[peteratcam]

Forget the QM part, it isn't actually important. Your question is more a standard one from sampling theory - if I draw n samples from a distribution, how do the properties of the sample converge on the distribution as n gets very big. The convergence phrases you refer to in your post are more mathematical, different from this empirical effect where a histogram of a large number of samples from a distribution converges on that distribution.

 

[LarryS]

Forget the QM part, it isn't actually important. Your question is more a standard one from sampling theory - if I draw n samples from a distribution, how do the properties of the sample converge on the distribution as n gets very big. The convergence phrases you refer to in your post are more mathematical, different from this empirical effect where a histogram of a large number of samples from a distribution converges on that distribution.

I realize my question is more mathematical than physical/empirical. But much has been written about the purely mathematical aspects of QM (Von Neumann, etc.).

So, has anyone written about the very specific type of PDFs that are produced by quantum processes?