Imagine a neutron and an electron bound to each other by...

The gravitational force provides the necessary centripetal force for the electron to revolve around the neutron.

$\dfrac{G m_n m_e}{r^2} = \dfrac{m_e v^2}{r}$

$v^2 = \dfrac{G m_n}{r}$

According to Bohr's quantization rule for angular momentum:

$m_e v r = \dfrac{n h}{2 \pi}$

$r = \dfrac{n h}{2 \pi m_e v}$

Substituting $r$ in the expression for $v^2$:

$v^2 = \dfrac{G m_n 2 \pi m_e v}{n h}$

$v = \dfrac{2 \pi G m_n m_e}{n h}$

The total energy of the system is the sum of kinetic and potential energy, which is equal to the negative of the kinetic energy:

$E = -K = -\dfrac{1}{2} m_e v^2$

$E = -\dfrac{1}{2} m_e \left( \dfrac{2 \pi G m_n m_e}{n h} \right)^2$

$E = -\dfrac{2 \pi^2 G^2 m_n^2 m_e^3}{n^2 h^2}$