How does thermal wavelength work exactly? In many sources it is stated that the thermal wavelenth indicates the rough size of the atom. It is then stated that this wavelenght is the de-Broglie wavelength of a particle with a momentum with the average kinetic energy in that temperature.
I don't really understand what this means in the context of quantum theory. If a particle has a very well defined momentum value, its position is very well undefined. In the limiting case if the particle actually has a de-Broglie wavelength of any value, $\psi$ will be a complex sinusoid with no size at all.
Is the thermal wavelength just an observational fact? Have people just noticed that gas atoms interact with the environment in such a way that the atoms localize to an area which happens to be  the same as the wavelength of a momentum eigenstate with the eigenvalue of expected value of the atoms momentum? Or is there some proof for this somewhere?
 A: 
In many sources it is stated that the thermal wavelenth indicates the rough size of the atom. It is then stated that this wavelenght is the de-Broglie wavelength of a particle with a momentum with the average kinetic energy in that temperature.

Average kinetic energy of an atom is
$$\frac{3}{2}k_B T$$
de-Broglie wave is not an arbitrary wave function $\psi$, but a plane wave with energy
$$ E=\hbar\omega=\frac{\hbar^2 k^2}{2m},$$
where $k=\frac{2\pi}{\lambda}$ is the wave vector.
Thus, the definition requires us to equate these quantities:
$$
\frac{3}{2}k_B T = \frac{\hbar^2 k^2}{2m}
= \frac{h^2}{2m\lambda^2} \Rightarrow \lambda=...$$
Rough size of atom
Note that here we treat an atom as a point-like particle, described by a De-Broglie wave. That is, the wave function here is for the overall motion of the atomic center-of-mass, rather than for an inner state of an atom, which is extensively discussed in QM books. This might be a possible source of confusion. Thus, what is meant here by the size of atom is not the size of the electronic cloud around the nucleus, but rather the extension of the wave packet in space.
