The binding energy of a H-atom, considering an electron moving around a fixed nuclei (proton), is $\mathrm{B}=-\frac{{\mathrm{me}}^4}{8{\mathrm{n}}^2{\mathrm{\epsilon }}_0^2{\mathrm{h}}^2}(\mathrm{m}=\text{ electron mass })$. lf one decides to work in a frame of reference where the electron s at rest, the proton would be moving around it. By similar arguments, the binding energy would be $\mathrm{B}=-\frac{{\mathrm{Me}}^4}{8{\mathrm{n}}^2{\mathrm{\epsilon }}_0^2{\mathrm{h}}^2}(\mathrm{M}=\text{ proton mass })$.
This last expression is not correct because
 

ln a hydrogen atom, electrons revolving around a fixed proton nucleus have some centripetal acceleration. So that, its frame of reference is non-inertial. ln the frame of
reference, where the electron is at rest, the Biven expression is not true as it forms the non-inertial frame of reference.
As the mass of an electron is negligible as compared to proton, so the centripetal force cannot provide t}le electrostatic force,
${\mathbf{F}}_{\mathrm{p}}=\frac{{\mathrm{m}}_{\mathrm{p}}{\mathrm{v}}^2}{\mathrm{r}}$
So the given expression is not true, as it forms non-inertial frame of reference due to ${\mathrm{m}}_{\mathrm{e}}<<<{\mathrm{m}}_{\mathrm{p}}$or centripetal force on ${\mathrm{F}}_{\mathrm{e}}<<<{\mathrm{F}}_{\mathrm{p}}$.