PHYS 424 Notes

There is one way to get in trouble: if E_n⁽⁰⁾ = E_m⁽⁰⁾ for any combination of n and m. In that case |<m|H'|n>|² / (E_n⁽⁰⁾ - E_m⁽⁰⁾) is not ordinarily smaller than <n|H'|n>! So the whole theory (so far) fails if the states of the unperturbed Hamiltonian are degenerate.

The cure is to ensure that <m|H'|n> = 0 whenever E_n⁽⁰⁾ = E_m⁽⁰⁾. Then there is no problem with infinities. This is possible and legitimate because any basis within the space of degenerate states is a legitimate set of unperturbed states. So anytime you are going to do perturbation theory where the unperturbed states are degenerate, first diagonalize the perturbing Hamiltonian. Note that the diagonalization is necessary even if you are not going beyond first order.

1. Source of the energy
   • Magnetic field of the proton in the electron rest-frame
   • Magnetic moment of electron
   • Thomas precession effect
2. H' = [e²/(4pe₀)] [2m²c²r³]^-1 L^.S
3. Choice of L=1 states [L|00>=0]
4. [L, L^.S] = i hbar S x L
5. [S, L^.S] = i hbar L x S
6. [J, L^.S] = 0
7. [L²,L^.S] = 0
8. [S²,L^.S] = 0
9. For states n=2, l=1
   1. H'_nm in natural states
      1. L^.S = L_zS_z + (L₊S_- + L_-S₊)/2
      2. States |M_Lm_s>:
         1: | 1 +1/2>
         2: | 1 -1/2>
         3: | 0 +1/2>
         4: | 0 -1/2>
         5: |-1 +1/2>
         6: |-1 -1/2>
         
      3. Matrix

         1/2	0	0	0	0	0
         0	-1/2	sqrt(2)/2	0	0	0
         0	sqrt(2)/2	0	0	0	0
         0	0	0	0	sqrt(2)/2	0
         0	0	0	sqrt(2)/2	-1/2	0
         0	0	0	0	0	1/2

   2. H'_nm in JM_J
      1. L^.S = (1/2)[J² - L² - S²]
      2. <J'M'|L^.S|JM> = (hbar²/2) [J(J+1)-l(l+1) -s(s+1)] d_JJ' d_MM'
      3. <1/r³> = ò (1/6) a^-3 [r/(2a)]² e ^-r/a (1/r³) r² dr
             = (24a⁵)^-1 ò r e ^-r/a dr
             = (24a³)^-1
      4. E⁽¹⁾ = [e²/(4pe₀)] [hbar²/(96m²c²a³)] [j(j+1)-11/4]    

           m        e2      1
En = - [ -------  (-----)2] --
         2 hbar2   4p e0     n2

                1
   = - a2 m c2 ----
               2 n2