Editing Normal distribution (section)

=== Exact normality ===
[[File:QHarmonicOscillator.png|thumb|The ground state of a [[quantum harmonic oscillator]] has the Gaussian distribution.]]
A normal distribution occurs in some [[physical theory|physical theories]]:
* The [[Maxwell–Boltzmann distribution#Distribution for the velocity vector|velocity distribution]] of independently moving and perfectly elastic spheres, which is a consequence of [[Maxwell's theorem|Maxwell's Dynamical Theory of Gases, Part I (1860)]].{{sfnp|Maxwell|1860|p=23}}
* The [[ground state]] [[wave function]] in [[Position and momentum spaces#Quantum mechanics|position space]] of the [[quantum harmonic oscillator]].<ref>{{cite book |last1=Larkoski |first1=Andrew J. |title=Quantum Mechanics: A Mathematical Introduction |date=2023 |publisher=Cambridge University Press |location=United Kingdom |pages=120-121 |url=https://www.google.com/books/edition/Quantum_Mechanics/iKmnEAAAQBAJ?hl=en&gbpv=1&dq=normal%20distribution&pg=PA120&printsec=frontcover |access-date=30 May 2025}}</ref>
* The position of a particle that experiences [[diffusion]]. If initially the particle is located at a specific point (that is its probability distribution is the [[Dirac delta function]]), then after time ''t'' its location is described by a normal distribution with variance ''t'', which satisfies the [[diffusion equation]]&nbsp;<math display=inline>\frac{\partial}{\partial t} f(x,t) = \frac{1}{2} \frac{\partial^2}{\partial x^2} f(x,t)</math>. If the initial location is given by a certain density function <math display=inline>g(x)</math>, then the density at time ''t'' is the [[convolution]] of ''g'' and the normal probability density function.