# Bound states between neutrinos using Schrödinger's equation?

I would like to see if it's possible that neutrinos (with sufficiently slow velocities) could form bound states in a universe with matter (such as ours)

There is a cosmic neutrino background in the universe which is probably the coldest neutrinos we could find (meaning slowest speeds). Slower neutrinos would heat up by thermalizing to the background.

I have been told that their wave function is around 1cm in length at those temperatures. I would think there is simply too much other matter around for the neutrino to form some sort of bound gravitational state but I'm not sure.

However I've read that I could plug this into Schrödinger’s equation and replace the potential term with the gravitational potential and see what the bound states energy are (I'd have to guess at the mass of the neutrinos, but I could use some upper limit for your guess)

However, I'm studying physics by my own (as I come from another scientific field that is not very related with it) and I have a bit of trouble understanding Schrödinger's equation

I don't think this would be difficult, but could someone help me how to figure this out? How can I solve the equation for this? Which steps should I take?

• If you use the Newtonian potential, then I think this is the same solution as for the Hydrogen-like atoms. The possible difference lies in the interpretations of the counterpart of the Bohr radius. However, note that neutrinos have 1/2 spin, so the Dirac equation is better suited for such a calculation. Commented Jun 20 at 14:31
• Commented Jun 20 at 17:26

## 1 Answer

A quick back-of-the envelope estimate, in the style of Fermi:

For a neutrino mass of m ~ 1 eV, and a Planck mass M ~ $$10^{27}$$ eV, and supplanting the newtonian potential $$(m/M)^2/r$$ for the Coulomb potential $$e^2/r$$, yields a Bohr radius for such neutrinuum atoms of $$r \sim M^2/m^3 \sim 10^{54}/\hbox{eV} \sim 10^{47} ~~\hbox{m}.$$
Compare this to the present diameter, $$10^{27}$$ m, of the entire universe...

• So we have to just wait long enough ... Commented Jun 21 at 12:44
• @CosmasZachos in the question that you linked (physics.stackexchange.com/questions/27498/…) the OP asks after getting the answers: "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?". I have basically the same question now, if not now, would neutrinos be able to form bound states once everything in the universe has cooled down? Commented Jun 23 at 10:19
• My point above is that such bound states don’t fit in the universe now or in the conceivable universe in future, so the are irrelevant; not that they are so weakly bound so as to be unstable… Commented Jun 23 at 10:30
• @CosmasZachos two questions: 1# when you say "in the conceivable universe in future" what scale of years are you referring to? 2# then the conclusion is that these bound states are impossible no matter how far we go into the future of the universe? Commented Jun 25 at 9:59
• 2. Indeed, because, 1. Compare todays size with its expansion rate… Commented Jun 25 at 10:32