Question

9. An electron moving with non-relativistic velocity v in an electric field E experiences a magnetic fieldB given by: v x (-V(r)) v x E B=- where (r) is the electric potential. This magnetic field interacts with the magnetic moment u of the electron given by -S, =n me where S is the electron spin. Assuming non-relativistic mechanics, show that the Hamiltonian representing this effect (spin-orbit coupling) for a spherically-symmetric electric potential is: 1 dφ(r) S.L ΔΗ [6] r dr mec2 Because of Thomas precession, AH is divided by a factor of two. Taking only the effect of the spin-orbit coupling into account in the hydrogen atom, calculate the energy splitting of the n = 2 excited state and the n = 3 excited state assuming that the electron could be in any of the allowed eigenstates of orbital angular momentum. Draw a diagram clearly showing the splittings and label the energy levels using spectroscopic notation [10

Answer 1

alassMate Date Page 8- mc m, c2 E(e V(y - ield eluchon the nuclus alulatrd hn th sest Thi abtalnid du ixp empt o d em, &ds in tegum m, dr and it Me -e AH +esB me tnterachton the Themas peuceson Comsidong mut diud AH= t derr) v e dr Vor)- A 2 73

SMACE Date Page <nljnlisnjm) 2 ) 2 e it) )- (1) THST(HT e2 137 2 ans me 8n3 (TT(TPT

Page DEse Gt Pe -1 03 epcahan kially ablkm Note thet tais ingular tat o ond it s nat hile value barause const Const 3P A&o Const camat Cet ab Canst 1-2 7 7 -1 -0614 X1X 3 3 Dstェ S 27x5x3 0 27KA 210(S)

Aa 0-013 97 X 2+ 2 z 0.024 2932 bchr leul 114a, 2 P