Q28 of 29 Page 79

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


=


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If a proton had a radius R and the charge was uniformly distributed, calculate using Bohr theory, the ground state energy of a H-atom when (i) R = 0.1Å, and (ii) R = 10 Å.

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In the Auger process an atom makes a transition to a lower state without emitting a photon. The excess energy is transferred to an outer electron which may be ejected by the atom. (This is called an Auger electron). Assuming the nucleus to be massive, calculate the kinetic energy of an n = 4 Auger electron emitted by Chromium by absorbing the energy from a n = 2 to n = 1 transition.

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The Bohr model for the H-atom relies on the Coulomb’s law of electrostatics. Coulomb’s law has not directly been verified for very short distances of the order of angstroms. Supposing Coulomb’s law between two opposite charge +q1, –q2 is modified to



Calculate in such a case, the ground state energy of a H-atom, if ε = 0.1, R0 = 1Å.