The inverse square law in electrostatics is for the force between an electron and a proton. The dependence of |F| can be understood in quantum theory as being due to the fact that the ‘particle’ of light (photon) is massless. If photons had a mass mp, force would be modified to where and .

Estimate the change in the ground state energy of a H-atom if mp were 10–6 times the mass of an electron.


Wavelength




∴ --------(1)


Now equating the new force equation with centripetal force,


------(2)


Where r is the new radius.


But we know that, ,putting this in equation 1


----------(3)


For , we get Bohr radius of the electron, ------(4)


Now, let the change in the orbital radius be p which is a small change.


∴ , ------(5)


∵ eq. 3 and 2 are the same, we equate them,



Now p2 and prb are ≪1


∴ neglecting them, --------(6)


Now kinetic energy


K.E. =


Now putting value of p and ,K.E for hydrogen atom ground state


∴ K.E


Now P.E. ----------(7)


Let


Differentiating w.r.t r



Substituting above in the integral (7) we get,


P.E.




Using equation (5) and putting , we get,




Using equation (6) ()




Now putting the values, we get,



(using )


∴ change in ground state energy


=


1