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Apart from the wave function in a harmonic well. Are all discovered normal distribution in the physical world a result of central limit theorem?

If it is the case, it may allow some reverse reasoning when we see another normal distribution. Though I guess it is already applied in some domain.

I didn't think it thoroughly, so it may be a nonsense question.

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  • $\begingroup$ According to Cunningham’s Law, it may be more effective to assert that all natural normal distributions arise from the CLT and see if anyone pushes back with a counterexample. $\endgroup$ – Chemomechanics Mar 31 at 16:16
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There are several ways to motivate measurement errors being Gaussian (Jaynes has a whole chapter on them, summarised here).

Normal distributions are found elsewhere too. jacob1729 noted one example of a Normal distribution resulting (in thermal equilibrium) from a quadratic energy, a very important scenario. Another interesting example is a quantum SHO's ground state, which is Normal in either $x$ or $p$-space; the reason is we have to solve $\hat{a}|\psi\rangle=0$.

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The $x$ component of the velocity of an ideal gas is normally distributed for entirely different reasons, so no.

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  • $\begingroup$ That distribution can’t be explained by applying the CLT to a large number of random kicks from previous collisions? $\endgroup$ – Chemomechanics Mar 31 at 20:42
  • $\begingroup$ @Chemomechanics if you can find a sensible way to make that work I'd be interested. It's possible but the barriers I see are: (1) each collision essentially randomises the velocities of both atoms which makes the "many collisions" idea seem wrong and (2) why should that apply to components and not to the speed? $\endgroup$ – jacob1729 Mar 31 at 20:56
  • $\begingroup$ @Chemomechanics No. If you had a gas of massless particles, the velocity is distributed like $e^{-|v|}$, but your logic would say it's also $e^{-v^2}$. $\endgroup$ – knzhou Apr 1 at 16:43
  • $\begingroup$ For a relativistic treatment, see here. $\endgroup$ – J.G. Apr 1 at 18:20

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