# Does Nature Like Gaussians?

1. Jun 17, 2010

### referframe

A Gaussian function is often chosen as the amplitude function for the wave packet of a single particle. It is handy because the Fourier Transform of a Gaussian is also a Gaussian. So the position and momentum representations would both be Gaussians.

But is there any empirical evidence that nature prefers Gaussians for single-particle wave packets?

2. Jun 17, 2010

### Zarqon

The shape of things are usually related to the statistical rules under which they form.

For example, for linewidths, the shape of the natural linewidth is obtained from the underlying rule that an atom has a uniform probability per unit time to decay, i.e. it follow poisson statistics. From this the Lorentizan lineshape follows automatically.
When a line instead is not limited by decay, but for example, as is very common, by doppler broadening, then the shape changes. Doppler broadening is an inhomogeneous effect, i.e. each photon may be narrow, as all are different they sum up to a broader shape following the normal distribution, and from this it follows that the line now has a Gaussian shape.

Infact, as far as I know, inhomogeneities usually gives rise to gaussians.

This may not be exactly what you asked, but I think that if it's a wavepacket that you create somehow, then you have the freedom to also select it's shape, and experimentally it is possible to create gaussian wavepackets, so there's certainly no problem with using them in the theory if it makes it simpler.

3. Jun 17, 2010

### alxm

Gaussians minimize the uncertainty relationship. I.e. $$\Delta x \Delta p = \hbar$$ for gaussians.