Gaussian

The Uncertainty Principle states that,

From quantum mechanics, gaussians are the most “certain” wave functions. (1) The “Heisenberg uncertainty principle” states that for any wave function, where 'delta x' is the uncertainty in position and 'delta p' is the uncertainty in momentum. ℏ represents h/2π, where h denotes Planck's constant: 6.626 × 10-34 m2 kg / s

For a gaussian, this equation is the absolute minimum total uncertainty. It is mentioned from many sources that the probability distribution of the particle position and momentum would follow a Gaussian distribution.

Notes:

Why is it a Gaussian distribution? is this the distribution that minimizes uncertainty? Is this distribution definitely the case for the uncertainty principle or can it be different under different conditions? Has this been proven?

What are the formulas for position and momentum probability distributions of a free particle? How is this derived from the wave function? What would be the formulas of the probability distributions for the position and momentum for a system of 2 identical bosons separated by a distance RR?

(1). http://www.askamathematician.com/