ch12 - CHAPTER 12 1 With a potential of the form V(r 1 m 2...

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CHAPTER 12. 1. With a potential of the form V ( r ) = 1 2 m ϖ 2 r 2 the perturbation reduces to H 1 = 1 2 m 2 c 2 S L 1 r dV ( r ) dr = ϖ 2 4 mc 2 ( J 2 - L 2 - S 2 ) = ( h ϖ ) 2 4 mc 2 j ( j + 1 ) - l ( l + 1 ) - s ( s + 1 ) ( 29 where l is the orbital angular momentum, s is the spin of the particle in the well (e.g. 1/2 for an electron or a nucleon) and j is the total angular momentum. The possible values of j are l + s , l + s – 1, l + s –2, …| l s |. The unperturbed energy spectrum is given by E n r l = h ϖ (2 n r + l + 3 2 ) . Each of the levels characterized by l is (2 l + 1)-fold degenerate, but there is an additional degeneracy, not unlike that appearing in hydrogen. For example n r =2, l = 0. n r =1, l = 2 , n r = 0, l = 4 all have the same energy. A picture of the levels and their spin-orbit splitting is given below. 2. The effects that enter into the energy levels corresponding to n = 2, are (I) the basic Coulomb interaction, (ii) relativistic and spin-orbit effects, and (iii) the hyperfine structure which we are instructed to ignore. Thus, in the absence of a magnetic field, the levels under the influence of the Coulomb potential consist of 2 n 2 = 8 degenerate levels. Two of the levels are associated with l = 0 (spin up and spin down) and six
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levels with l = 0, corresponding to m l = 1,0,-1, spin up and spin down. The latter can be rearranged into states characterized by J 2 , L 2 and J z . There are two levels characterized by j = l –1/2 = 1/2 and four levels with j = l + 1/2 = 3/2. These energies are split by relativistic effects and spin-orbit coupling, as given in Eq. (12-16). We ignore reduced mass effects (other than in the original Coulomb energies). We therefore have E =- 1 2 m e c 2 α 4 1 n 3 1 j + 1/2 - 3 4 n =- 1 2 m e c 2 α 4 5 64 j = 1/2 =- 1 2 m e c 2 α 4 1 64 j = 3/2 (b) The Zeeman splittings for a given j are E B = e h B 2 m e m j 2 3 j = 1/2 = e h B 2 m e m j 4 3 j = 3/2 Numerically 1 128 m e c 2 α 4 1 .132 × 10 - 5 eV , while for B = 2.5T e h B 2 m e = 14 .47 × 10 - 5 eV , so under these circumstances the magnetic effects are a factor of 13 larger than the relativistic effects. Under these circumstances one could neglect these and use Eq. (12-
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