CHAPTER 6: AN APPLICATION OF PERTURBATION THEORY THE FINE AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM. (From Cohen-Tannoudji, Chapter XII)

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1 CHAPTER 6: AN APPLICATION OF PERTURBATION THEORY THE FINE AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM (From Cohen-Tannoudji, Chapter XII)

2 We will now incorporate a weak relativistic effects as perturbation of the non-relativistic Hamiltonian Ĥ = Ĥ 0 + W ˆ P 2 ˆ P 4 = m e c V(R) } 2m e {{ }} 8m {{ 3 ec } 2 Ĥ 0 Ŵ mv dv(r) ˆ L 2m 2 ec 2 ˆ S R dr } {{ } Ŵ S O + 2 } 8m 2 ec {{ 2 V(R) +...(6.1) } Ŵ D Ŵ mv - variation of mass with velocity Ŵ S O - spin-orbit coupling Ŵ D - Darwin term

3 The energies relevant to the relativistic effects are weak compared to the energy associated with Ĥ 0 Ŵmv Ĥ 0 ŴS O Ĥ 0 ŴD Ĥ 0 Ŵmv Ĥ 0 α 2 ( ) 1 2 (6.2) 137 In addition we will consider hyperfine structure which comes from the interaction of the electron and nuclear magnetic momenta Ŵ h f = µ { 0 q ˆ L 4π m e R 3 ˆ M I + 1 [ ( ( ) R 3 3 ˆ M S ˆn) ˆ M I ˆn ˆ M ] S ˆ M I (6.3) + 8π 3 ˆ M S ˆ M I δ ( R )} ŴS O 2000 (6.4) ˆ M I = g Pµ nˆ I, µ n = q P (6.5) 2M P

4 Fine structure of the n = 2 level 2s (n = 2, l = 0) and 2p (n = 2, l = 1) E = E I 4 = 1 8 µc2 α 2 (6.6) Orbital angular momentum ˆL z : l = 0, m = 0 and l = 1, m = +1, 0, 1 Electron spin Ŝ z : m = ±1/2 Nuclear spin Î z : m = ±1/2 The perturbation Hamiltonian Ŵ f = Ŵ mv + Ŵ S O + Ŵ D (6.7) Ŵ = Ŵ f + Ŵ h f (6.8)

5 Matrix representation of Ŵ f matrix Ŵ f does not act on ths spin variables of the proton Ŵ f does not connect the 2s and 2p subshells ˆL 2 commutes with Ŵ f

6 1. 2s subshell m s = ±1/2 2 dimensional Ŵ mv and Ŵ D do not depend on ˆ S they are proportional to a unit matrix and are given as Ŵmv 2s ŴD 2s = n = 2; l = 0, m L = 0 ˆ P 4 8m 3 ec 2 n = 2; l = 0, m L = 0 = m ec 2 α 4 (6.9) 2 = n = 2; l = 0, m L = 0 8m 2 ec 2 V(R) n = 2; l = 0, m L = 0 = 1 16 m ec 2 α 4 (6.10)

7 and since l = 0 ŴS O = 0 (6.11) Thus the fine structure terms lead to shifting the 2s subshell as a whole by an amount 5m e c Ŵmv + ŴD + ŴS 2 α 4 O = (6.12) 128

8 2. 2p subshell (a) Ŵ mv and Ŵ D terms commute with ˆ L and do not act on spin ˆ S variables a multiple of a unit matrix (See Complement BXII) Ŵmv 2p = m ec 2 α 4 (6.13) ŴD 2p = 0 (6.14)

9 (b) Ŵ S O various elements: n = 2; l = 1; s = 1 2 ; m L ; m S ξ(r) ˆ L ˆ S n = 2; l = 1; s = 1 2 ; m L; m S ξ(r) = e2 1 2m 2 ec 2 R 3 (6.15) In { r } representation, we can separate the radial part of the matrix elements from the angular and spin parts: ξ 2p l = 1; s = 1 2 ; m L ; m S ˆ L ˆ S l = 1; s = 1 2 ; m L; m S (6.16) ξ 2p = e2 2m 2 ec 2 Problem: the diagonalization of the ξ 2p ˆ L ˆ S operator 0 1 r 3 R 21(r) 2 r 2 dr = m ec 2 α 4 (6.17)

10 Problem: the diagonalization of the ξ 2p ˆ L ˆ S operator Basis { l = 1; s = 1/2; m L ; m S } common eigenstates of ˆL 2, Ŝ 2, ˆL z, Ŝ z Introducing the total angular momentum ˆ J = ˆ L + ˆ S (6.18) { l = 1; s = 1 } 2 ; J, m J (6.19)

11 Addition of angular momentum: J = 1 + 1/2 = 3/2 and J = 1 1/2 = 1/2 ( Jˆ 2 = ˆ L + ˆ S ) 2 = ˆL 2 + Ŝ ˆ L ˆ S (6.20) then ξ 2p ˆ L ˆ S = 1 2 ξ 2p ( ˆ J 2 ˆL 2 Ŝ 2) (6.21) ξ 2p ˆ L ˆ S l = 1; s = 1 2 ; J, m J = 1 [ 2 ξ 2p 2 J(J + 1) 2 3 ] l = 1; s = ; J, m J (6.22)

12 The eigenvalues of ξ 2p ˆ L ˆ S depend only on J and not on m J, and are equal to J = 1 [ 2 : ξ 2p J = 3 2 : 1 2 ξ 2p ] 2 = ξ 4 2p 2 = 1 48 m ec 2 α 4 (6.23) [ ] 2 = ξ 2p 2 = 1 96 m ec 2 α 4 (6.24) The six-fold degeneracy of the 2p level is therefore partially removed by Ŵ S O

13 The fine structure of the n = 2 level: energies 2s 1/ m ec 2 α 4 (6.25) 2p 1/2 ( ) m e c 2 α 4 = m ec 2 α 4 (6.26) 2p 3/2 ( ) m e c 2 α 4 = m ec 2 α 4 (6.27)

14 The fine structure of the n = 2 level

15 The hyperfine structure of the n = 1 level a) The degeneracy of the 1s level no orbital degeneracy (l = 0) Ŝ z, Î z : each 2 value ±1/2 { n = 1; l = 0; m L = 0; m S = ± 12 ; m I = ± 12 } (6.28)

16 b) The 1s level has no fine structure Ŵ f does not remove the degeneracy of the 1s state Ŵmv ŴD ŴS O 1s 1s 1s = 5 8 m ec 2 α 4 (6.29) = 1 2 m ec 2 α 4 (6.30) = 0 (6.31) Ŵ f shifts the levels by 1 8 m ec 2 α 4 Since l = 0 and s = 1/2, J = 1/2 only one fine structure level, 1s 1/2

17 Matrix representation of Ŵ h f in the 1s level Ŵ h f = µ 0 4π q ˆ L m e R 3 ˆ M I } {{ } =0 as l=0 + 8π ˆ M 3 S ˆ M I δ ( R ) } {{ } ( ( ) [3 ˆ M R 3 S ˆn) ˆ M I ˆn ˆ M I ] ˆ M S } {{ } =0 due to spherical symmetry of 1s state (6.32) (6.33) ˆ M I = g Pµ nˆ I, µ n = q P (6.34) 2M P

18 The matrix elements are n = 1; l = 0; m L = 0; m s; m I 2µ 0 3 ˆ M S ˆ M I δ ( R ) n = 1; l = 0; m L = 0; m s ; m I In coordinate representation, we separate the orbital and spin parts of the matrix elements A m S, m I ˆ I ˆ S m S, m I (6.35) where A = = q 2 g P 3ɛ 0 c 2 n = 1; l = 0; m m e M L = 0 δ ( R ) n = 1; l = 0; m L = 0 (6.36) P q 2 g P 1 3ɛ 0 c 2 R m e M P 4π 10 (0) 2 4 = 3 g m ( e p m e c 2 α m ) 3 e 1 M P M P 2 (6.37)

19 The orbital variables have thus disappeared, and we are left with the problem of two spin 1/2 s, coupled by an interaction Aˆ I ˆ S (6.38) where A is a constant. Eigenstates and eigenvalues Initial basis { s = 1/2; I = 1/2; m s ; m I } common eigenvectors of Ŝ 2, Î 2, Ŝ z, Î z New basis introducing total angular momentum ˆ F = ˆ S + ˆ I: { s = 1/2; I = 1/2; F; m F } common eigenvectors of Ŝ 2, Î 2, ˆF 2, ˆF z

20 ˆ F: F = 0, m F = 0 and F = 1, m F = 1, 0, 1 Aˆ I ˆ S = A 2 ( ˆF 2 Î 2 Ŝ 2) (6.39) The basis states F, m F are eigenstates of Aˆ I ˆ S : Aˆ I ˆ S F, m F = A 2 2 [F(F + 1) I(I + 1) S (S + 1)] F, m F (6.40) F = 1 : F = 0 : A 2 [ ] = A 2 (3-fold degenerate) (6.41) 4 4 A 2 [ ] = 3A 2 (nondegenerate) (6.42) 4 4 The four-fold degeneracy of the 1s level is partially removed by Ŵ h f.

21 The hyperfine structure of the 1s level

22 The n = 2 level