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RE : Second exercise
posted by Mathieu Gisselbrecht at Feb 14, 2012 7:48 PM
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Thanks for the reformulation of the slide about the use of the LS and jj coupling !… They are a few books dealing with this with more or less explanation. In any case, books will provide more information that one can give in two hours ! A deep course on atomic and molecular structure can take up to six-months.

The book of Brandsden & Joachain is a relatively good one. As it was pointed out you have the solution of these "textbook" examples, which was the meaning of the exercises: learn how to couple different angular momentums for different basis and make the connexion with the notation, i.e learn how to read them.

If someone wants to go deep in details into all the quantum mechanic aspect of the problem, I personally recommend: R.D. Cowan, The theory of the atomic structure and spectra

RE : Second exercise
posted by fyst20 at Feb 13, 2012 6:35 PM
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So there have been a couple of people today at my office asking about the difference of LS and JJ coupling.

And apparently my Notes were not sufficient to clear that up, so i will give it a second try here.

Postulate 1 The Multiparticle Hamiltonian can be written as H = sum_i ( p^2/2m + V'(r) + H'' ) - Central Field Approximation -

Conclusion The Multiparticle Wavefunctions expanded in the |n l ml s ms > Basis (Spherical Harmonics) trivially diagonalizes our Hamiltonian, (are good quantum numbers see Foot, Chapter 2), as long as H'' is approx. 0 (meaning that the central field approximation is valid)

Postulate 2 The Spin Orbit interaction (Combination of Thomas Interaction and Larmor Precession) can be expressed as H_s_o = beta * L scalar S. It follows the Hamiltonian H = H0 + H_s_o

Conclusion In first order perturbation theory (H_s_o << H') there is a splitting in J (see Foot. p.83, 84) (1)

Postulate 3 The H_s_o term becomes comparable to the rest of the Hamiltonian at high Z numbers.

Conclusion The first order perturbation approach is no longer valid. A trivial solution of the problem is also no longer possible (as far as i know). (2)

Baseline: In region (1) the effect is called LS splitting in region (2) the effect is called JJ splitting. LS splitting happens in J. Whereas JJ splitting happens in mj (without proof, see Foot or Brandsden & Joachain).

If you are wondering where to find a proper description of the notation. I have not found one. If you are wondering where to find a description that allows you to solve:

  • the first question - Brandsden & Joachain (p.415)
  • the second question - Brandsden & Joachain (p.408)
  • the third question - No idea. Couldnt find a proper derivation of the Jcl scheme.
Second exercise
posted by Mathieu Gisselbrecht at Feb 9, 2012 10:51 AM
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