Consider a neutrom and an electron bound to each other due to gravitational force. Assuming Bohr's quantization rule angular momentum to be valid in this case, derive an expression for the energy of the neutron-electron system.

Explanation

To move in circle, necessary centripetal force is provided by gravitational force.
`(m_(e)v^(2))/(r )=G(m_(e)m_(n))/(r^(2)) " " (i)`
`v=sqrt((Gm_(n))/(r ))`
Bohr's quantisation rule
`m_(e)vr=(nh)/(2pi) " " (ii)`
`m_(e)sqrt((Gm_(n))/(r ))r^(2)=(n^(2)h^(2))/(4pi^(2))`
Squaring both sides, we get
`m_(e)^(2)(Gm_(n))/(r ) r^(2)=(n^(2)h^(2))/(4pi^(2))`
`r=r_(n)=(n^(2)h^(2))/(4pi^(2)Gm_(e)^(2)m_(n))`
`K.E.: K=(1)/(2) m_(e)v^(2)=(Gm_(e)m_(n))/(2r)`
`P.E.: U=-(Gm_(e)m_(n))/(r )`
`T.E.: E=K+U=-(Gm_(e)m_(n))/(2r)`
`E_(n)=-(Gm_(e)m_(n))/(2xx(n^(2)h^(2))/(4pi^(2)Gm_(e)^(2)m_(n)))`
`=-(2pi^(2)Gm_(e)^(3)m_(n)^(2))/(n^(2)h^(2))`