Reference-http://www.ribbonfarm.com/2015/08/20/qft/

The basic indication was that there is no compelling reason to choose any particular shape,

although there will obviously be reasons for discarding obviously incorrect shapes.

Basically the Gaussian seems reasonable and sensible and is not obviously incorrect.

Even so I don't really see why they would be Gaussian, either. Gaussian wave functions arise in the harmonic oscillator problem and the quark-quark problem is anything but. At a bad approximation the quark potential is closer to a Yukawa potential than anything else and those get really messy.

The basic indication was that there is no compelling reason to choose any particular shape,

although there will obviously be reasons for discarding obviously incorrect shapes.

Basically the Gaussian seems reasonable and sensible and is not obviously incorrect.

-Dan

Any time we have a measurement of a physical quantity we can think of that as combining many "sub-quantities" and so expect that it will have, at least approximately, the normal distribution.

("At least approximately": if the population from which the samples are drawn is normal, both sum and average will have, exactly, the normal distribution. If the population is not normally distributed, the distribution of average and sum will approach normal as n goes to infinity.)

----------------anyprobability distribution, as long as the mean is \(\displaystyle \mu\) and the standard deviation is \(\displaystyle \sigma\), then the average of all the samples will be, at least approximately,normallydistributed with mean \(\displaystyle mu\) and \(\displaystyle \sigma\). The sum of n such samples will be, at least approximately, approximately normal with mean \(\displaystyle n\mu\) and standard deviation \(\displaystyle \sigma\sqrt{n}\).

Any time we have a measurement of a physical quantity we can think of that as combining many "sub-quantities" and so expect that it will have, at least approximately, the normal distribution.

("At least approximately": if the population from which the samples are drawn is normal, both sum and average will have, exactly, the normal distribution. If the population is not normally distributed, the distribution of average and sum will approach normal as n goes to infinity.)

The baseline Gaussian distribution is the result of actual measurements of position. This baseline was then improved with modifications to account for discrete quantum effects.

Yes, CLT can be used; but it is a best suited for situations where there is not an obvious standard distribution to fit the sample data. Images of sample particle positions reflect a clearly Gaussian-like pattern by inspection, and this natural fit to the bell-shaped curve is astonishing close!

Yes! Gaussian distributions are indicative of an underlying oscillations! This is the QM observation made that the exact position of a particle is difficult because of its continuous oscillations at a frequency that is part of the position state vector and determined by the momentum of the particle.

Don't confuse quark movement with that of the particle! When measuring particle movement, we treat the particle as a whole system. Movement within the system must necessary net to zero deviation from the observed particle states, including position, charge, and spin.