# 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

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...