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 m_p, force would be modified to where and .

Estimate the change in the ground state energy of a H-atom if m_p 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 pr_b 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

=