# Quantum mechanical gravitational bound states

The quantum mechanics of Coloumb-force bound states of atomic nuclei and electrons lead to the extremely rich theory of molecules. In particular, I think the richness of the theory is related to the large mass ratio between the nucleon and the electron. This mass ratio leads to the Born-Oppenheimer approximation which gives rise to a complicated effective potential for the nuclei which posses many local minima: the molecules.

I wonder whether there are analogical phenomena in which the bound states are gravitational. It seems that if we take a collection of electrically neutral molecules and neutrinos, it should be possible to form a large number of bound states, in particular because of the large (not so long ago deemed infinite) mass ratio between molecules and neutrions. Of course neutrinos are highly relativistic and I can't tell how it affects things.

Now, even if we leave neutrinos alone, the typical size of such a bound state is

$$\frac{\hbar^2}{G m^3}$$

where $m$ is proton mass. Google calculator reveals this to be 3.8 million light years. Holy moly! However, this is still much smaller than the observable universe. Can there be places in the universe sufficiently empty to contain such bound states? What would be the effect of general relativistic phenomena (expansion of space)? EDIT: I guess no place is sufficiently empty at least because background radiation is everywhere. Maybe these creatures will become relevant in a very distant future when the background radiation cools off considerably?

Summing up:

What is known about the quantum mechanics of gravitational bound states of electrically neutral molecules and neutrinos?

I'm tagging the question as "quantum gravity" since it involves gravity and quantum mechanics. Of course it is not quantum gravity in the usual sense of studying Planck-scale phenomena. I think the tag is still appropriate

EDIT: Gravitional bound states of molecules will often be unstable with respect to collapse to a van der Waals bound state (thx Vladimir for bring up the issue of vad der Waals interaction in the comments). However the lifetime of these states is very long.

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Sure but I'm interested in situations which are quantum mechanical (coherent) –  Squark Nov 12 '11 at 12:50
AFAIR there was a paper on eigenstates of cold atoms in (external) gravitational field. They were (again, AFAIR) hold by an evanescent laser beam. –  Piotr Migdal Nov 12 '11 at 13:07
@Vladimir, the Shroedinger equation is practically irrelevant for astronomic bodies because of decoherence. However, we may in principle consider chunks of molecular matter at zero temperature, provided sufficient isolation. The maximal relevant size such of such is a chunk is when the "Bohr radius" equals the size of the chunk, that is (hbar^2 / G * rho^3)^1/10. Typical density at zero temperature should be proton mass per (real) Bohr radius cubed, that is 10000 kg / m^3 (close to the density of lead). This leads to 100 nm, about the size of an HIV virus. –  Squark Nov 12 '11 at 13:59
And what is effect of the Van de Waals interaction of such chunks? –  Vladimir Kalitvianski Nov 12 '11 at 16:17