Lecture 12 Highlights One nice feature of Jris the fact that it is a “constant of the motion” for the perturbed Hamiltonian . Hence although we loose and as good quantum numbers, we gain soΗ+Η0lmsmjand . This allows us to write the spin-orbit perturbation operator jmLSrr•in terms of “constants of the motion” as (using the definition of SLJrrr+=, squaring it, and solving for LSrr•); ()22221SLJLS −−=•rr, so the eigenvalues of LSrr• are ())1()1()1(22+−+−+ ssjj llh, where for the electron. 2/1=s Finally we can calculate the first-order correction to the energy of the Hydrogen atom due to the spin-orbit perturbation. By the standard expression for first order energy correction: rdEnsonjsn30*01,,,ψψ∫∫∫Η=lAfter substituting the appropriate Hydrogen atom wavefunctions (that we shall derive later) and spin-orbit Hamiltonian one arrives at; )1)(21(24/3)1()1(201,,,++−+−+=lllllljjnEEnjsnα (here we are using the fact that s=1/2 for the electron.) Note that this energy shift is on the order of the fine structure constant squared times the unperturbed eigenenergy, exactly the same form as the relativistic correction. When the two results are combined (Homework 4) the result is the fine structure formula: ⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡+−=+=Δ21432201Re1jnnEEEEnSpinOrbitnlativitynα This correction is always negative and partially lifts the degeneracy of the unperturbed eigen-energies. See the schematic plot of the fine structure corrections to the energy levels of the Hydrogen atom elsewhere on the class web site. What is the connection between the total angular momentum Jrand our old friendsLr,Sr? If SLJrrr+= and has eigenvalue , has eigenvalue , and has eigenvalue , then one might naively think that 2J2)1( h+jj2L2)1( hll +2S2)1( h+sssj+=l. However this is not the whole story. If we apply to a souped-up Hydrogen atom product wave function which now includes spin,zJsmnmsmnmsYrRrs),()(),,(,,,,φθφθψlllll∝, where the spinor is denoted with the ket (s=1/2 for the electron), the result is: 1ssmsmnjmsmnzmJ,,,,,,,,llllhψψ=, and ()ssmsmnzzmsmnzSLJ,,,,,,,,llllψψ+= The operator sees only the spherical harmonic, while the operator sees only the spinor ket. The result is: zLzS ()ssmsmnsmsmnzmmJ,,,,,,,,lllllhhψψ+=, leading to the conclusion that sjmmm +=lHowever, if we operate with on this same Hydrogen atom wavefunction, the results are not so pretty. From the definition of 2JSLJrrr+=, we know that (note that SLSLJrr•++= 2222LrandSrcommute with each other), and the SLrr•operator can be expressed as: ()zzyyxxSLSLSLSL ++=• 22rr +−−+++= SLSLSLzz2. using our old friends the raising and lowering operators. can now be written as: 2J +−−+++++= SLSLSLSLJzz2222Now consider applying this operator to a hydrogen atom product wavefunction of the form 2121),()(),,(,,,,φθφθψlllllmnmsmnYrRrs= for example (i.e. with a “spin up” electron): ()( )2/1,2/1,1,,2/1,2/1,,,222/1,2/1,,,212243)1(−+++−+⎥⎦⎤⎢⎣⎡+++=lllllllllllhhhhhhllmnmnmnmmmJψψψ where we have used results from Eq. [4.121] and page 174 of Griffiths. Note that this is no longer an eigen-equation because the raising and lowering operators have changed the and values in the second term on the RHS. Apparently the eigenfunctions of are linear combinations of Hydrogen atom product wavefunctions lmsm2J),,(,,,,φθψrsmsmnll with different values of and . This is consistent with the statement we made in the last lecture that the lmsmSLrr•operator mixes together different unperturbed Hydrogen atom states. The eigenfunctions of can be expressed as linear combinations of states with different values of and using the world-famous Clebsch-Gordan coefficients ( ) as: 2JlmsmjsmmmjsCll smmmjsmmmjmsmCmjjsjs∑=+=lllll Where the ket ll mrepresents the spherical harmonics . The C-G coefficient values are given in Table 4.8 on page 188 of Griffiths. Remember that all of the coefficients should appear under a square root, with the minus sign (if any) out front. llmY
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