Question

2. (Total: 50 pts) a) (20 pts Show that for the hydrogen atom the sum of the relativistic first-order cor- rection where EC are the unperturbed energy levels in the hydrogen atom, and the first-order correction due to spin-orbit coupling so (+1)-1(1+1) - 8(8 + 1)] 1(1+3)(1+1) me2 gives the fine-structure formula (Hint: Think first about the possible relationships between j and l. How many do you have in case of the hydrogen atom? Then show that the fine-structure formula holds for each of these relations separately.]

Answer 1

The relativistic Kinectic energy of the electron is given by T = mc² - mc2 Now, the 1st term is the total relativistic energy and the and term is the rest energy we need to express Tin tems of the relativistic momentum. Þ mo vi- 2072 pet mg.minme*[1-(0)] - (0%) (t + met 1-% So Taj preto 4 - mc² nous expanding to powers of the e have small number (Ponc), wer

T= mc [] + (Gr 7 -1] - me a me Tits lon) - 5 (m-] amerélheted Chem.] 2m - 203 12t...n The loweet orden relatives tie Correction to the Hamiltonian is evidently. , p4 Er = (H) = {41/414) (pulpy Now, the Schrödinger equation (for the unperturbed States) says,

ply = 2m (E-v) 4 Herne, Elram -- (E-v)") [E-2E<V)+<3] 12h 4 2 2mc AS <tr> = Textanah <ty a na nra 2 mc {a eliminating a

.: The relativestic first order Correction is given by Where I Col is the unperfurbed energy levels in the hydrogen atom. The first order spén-orbit interaction tso =1&to). mirgo S.L. is Now, J = L+S .: J = (2+) (L+S) & +5+ 22.5 "L.S = 1 1 1 J L - 37

Therefore the eigen values of his are ť rj (5+1)-I(2+1) as 15+1)] d (d+ 2) (8+1) na e.tzz) [3(3+1) --(+1) -S(5+17 - 8 to.me & (&+/) (2 + 19 2² 23 s nj (j+1) - 2 (2+) -s (5+1)] } so - mc² 1 1 (0+1/2) (1+1) The fine – Structure correction is obtained by adding the expressions for the spins orbit and relativistic corrections is given by D r.40). {(+151(1+) -5 (5+1)? t mc (1+%) (2+1) ? 20m 2 (1+2

E5° ) ) ) - 35- (%-3) (از ) 28 = (1-3) ( h (+) -- (ع ) ( ز ) ا% - ز) 2mL ام در mc l=j+% hence j = &- or وو