Chapter 6 Methods of Approximation - Particle Physics Group

CHAPTER 6. METHODS OF APPROXIMATION 107Figure 6.2: Schematic diagram for the splitting of the 2p levels of a hydrogen atom as a functionof an external magnetic field B. For small B the degeneracy of the six 2p levels is completelyremoved, but as B becomes large the levels j = 3,m = 2 −1 and j = 1,m = 1 converge, so that2 2 2the degeneracy is only partially removed in the limit of very large B (Paschen-Back effect).C j m l ,m sm s = 1 2m s = − 1 2j = l + 1 2j = l − 1 2√l+m j +1/22l+1√l−m−j +1/22l+1√l−m j +1/22l+1√l+m j +1/22l+1with m j = m l + m s . For the two cases j = l ± 1 2 the matrix elements of S z thus become( 〈〈jlsm j |S z |jlsm j 〉 = C l±1 2jlsmm l ,+ 1 l , + 1 〈∣ + C l±1 2jlsm2 2m l ,− 1 l , − 1 )∣ ×2 2S z(C l±1 2∣m l∣jlsm,+ 1 l , + 1 〉+ C l±1 2∣2 2m l∣jlsm,− 1 l , − 12 2〉 ) (6.49)= ( ∣∣∣C l± 1 22m l∣ 2 ∣− ∣C l±1 ,+ 1 2m 2l∣ 2) = ± 2m j(6.50),− 1 2 2 2l + 1The energy shift induced by a weak external magnetic field is therefore∆Ejlm Z j= e⃗ [Bm j ± m ]j= eB2m e c 2l + 1 2m e c m j ·{ 2l+22l+1j = l + 1/22lj = l − 1/22l+1(6.51)The spin-orbit coupling already removes the degeneracy in j. From equation (6.51) we see thata weak external magnetic field in addition lifts the degeneracy in m j , thus explaining the namemagnatic quantum number. A level with given quantum numbers n and j thus splits into 2j +1distinct lines. As an example consider the 2p orbitals of the hydrogen atom. The 2p 3/2 level,with j = l + 1/2, splits into 4 levels according to m j = 3 2 , 1 2 , −1 2 , −3 2 with ∆EZ = eB2m ec m j · 4The 2p 1/2 levels split into two with m j = ±1/2 and ∆E Z = eB2m ec m j · 23(see figure 6.2).Strong field and Paschen–Back effect. In the case of very strong magnetic fields thespin-orbit term becomes (almost) irrelevant and the Zeeman term H Z forces the electrons intostates that are (almost) eigenstates of L z + 2S z = J z + S z . The total angular momentum J 2 is3 .