Calculating Binding Energy for a Neutron System with Gravitational Forces

[andre220]

Homework Statement

Consider a system of two neutrons interacting only through gravitational attractive forces. Find the binding energy of this "quantum atom" (in eV) and the characteristic size of the ground state configuration. Is there any chance to find such a system in reality?

Homework Equations

$$F = \frac{G m_1 m_2}{r^2}$$

The Attempt at a Solution

Okay so, I am not really sure where to start here. Obviously the force is as written above. My first though was to use $$U_G = -\frac{G m_1 m_2}{r}$$, but I am not really sure if this is on the right track, or, if so, where to go from there.

 

[clamtrox]

The first thing that comes to mind is that you can estimate the minimum kinetic energy from the Heisenberg uncertainty principle. For the neutrons to be bound, gravitational binding energy must be bigger than kinetic energy.

 

[TSny]

Hello, andre220. Welcome to PF.

The force of gravity between the two neutrons has exactly the same form as the electric force between the electron and the proton in a hydrogen atom. Only certain constants are different. So, recall the formula for the ground state energy of the hydrogen atom and figure out how to modify it for the gravity case.

There is an additional matter to consider. In the hydrogen atom, it is assumed to a good approximation that the proton is at rest and only the electron is moving. In the "neutron atom" both neutrons will be moving.