Question

Using the first order of perturbation theory, find the corrections to the ground state (1S) and state 2S of a hydrogen atom due to the finite size of the nucleus. Consider the nucleus be a sphere of radius R with uniformly distributed charge e.

Answer 1

Answer:

Perturbation theory --> Any of various methods of calculating the approximate value or complex function (such as the energy of an electron in quantum mechanics) by first assuming that the dominant influence is the only factor and then making small conections for additional factors. fine Shucture of Hydrogen atom:- for t-atom the hamiltonion defined as ho = 2 2 2 1 to e the this is energy. sum of é kinetic energy plus coulombic potential Fine Structure of H-atom consists two distinct mechanisms. Those are as folbws. (1) Relativistic Kinetic Energy Correction (ii) spin orbit coupling. 1) Relativistic Kinetic energy correction we know that the relativistic correction formula is . T= E-me 0 we know that - E=√ P²cam2c4 . equation @ Substitute in equation , we get 1 T = √6²2 27 m2 (4 _mc? miche en to] - med = me[1 + 12 7 ² - me? .
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= me[ part 1 ] - me = me [st z mi cele care of the T=p² - 14 .: am m? . So here, -p4 is a perturbation tem, then we neglect the sma2 is ao perturbation term, we get 2m then, He=-14 22 Perturbation tem. >3 In is this first order perturbation theory given by the expectation of H Ŕ in the correction of unperturbed state. En ER = = 2 <ulptu>=-1 <brul b²ws >$_14> = [amce-v > = 4 m["+<u?> -RE<v>] se ce est ent) as that has a < *3] —-0

: < >= a and < 1 2 > <ievadiety na Putting all the above equations in equation , we get Ce-ame [co +aco ) na tento roles) and we know that L a= 4T to 2² mez then, we get 2mclty () spin orbit.coupling : we know that, Hsome 8 TT to where Iso= spin orbit coupling Z=1+3 syuasing on both sides; we gel J* (J+) {=ML+)#+5(5+1) £127.5? 7:32 [3(+) -26C+) -s(5+1] <H:02 < > 25.

for one é, we know that <Hso>= e t t (5(J+1) - l(+1) -3/47 81 to me Lecl+D (1+2) na E = En n[J(J+1)_d(2+1)-3/4) leti) (l+/) Finally we know that fine structure & E = GES H-atom is Thus, we can findout the fine structure of H-atom by calculating the relcetivistic kinetic energy correction and spin onoit Coupling Now, by putting different value of n, we can find correction of any energy level