Question

Total: 24 pts] In our derivation of the first-order energy corrections for the spin-orbit coupling we have claimed that the unperturbed eigenstates of the hydrogen atom, {Inl misms >}, are not "good” wavefunctions to determine the first-order energy correction for the spin-orbit coupling e2 ) 1 H'so = (8160) m2c273 S.L. 8760 ) m a) (14 pts] To confirm our claim, calculate the commutators (S · L, L?) and (S · L, Lx]. What do you conclude? What do you expect, if you would replace Lº by S4 and L2 by Sz? You do not have to perform the calculations in the case of s? and Sz, but you should provide a brief expectation. b) [10 pts) Instead, we have used the set of simultaneous eigenfunctions to Ho, L², sa, J2 and Jz, which we denoted by {Inls j m; >}. Show that for a given n both sets, {Inl misms >} and {lnl s j m; >}, include the same number of eigenfunctions.

Answer 1

(c.o.M) finst onder energy Cormention for the spin-omobile Coupling. Ho = (475.) wein3.7). @ [$,2] - [Sii, e?] - Si[41, e?] * [S:,] [£3 2] = [L², L:g20 = 0 + (siegei 20... This [3.1 , 1². 120, 7 (22): conset of perfor similarly, [3.1, s²750 & (32com. [ 3.2, [2] = [5xlx + Syly & Salz , hz with Sy Lx it Sxly to (cond Duro) 7 izzi nof com. Simorty, [ 3.1, § 1 = wit Syly & ith Sylx to kan't rem.) (Sar: nof com. Now [1 + f2, 3 t 1 = 0 (het her: com. . Sut of simultancom eigenfunction to to, 1, 52, 7², Jz. - Os Lath. 92 1437 4 27.8 . Hof & Ho it to [02, Hot] all, 22 inligining & j (+1) 5 Inling> | 12 | nl mp3 myy a lleisk Ineme smy> I Intinza with the laj mj> freme smsy a menol nemes may This given in both sets {one mesme) and {m es ; m;>} , same Number of eigenfuntion.