notes25 - 5.73 Lecture #25 25 - 1 HSO + HZeeman Coupled vs....

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25 - 1 5.73 Lecture #25 revised 4 November, 2002 H SO + H Zeeman Coupled vs. Uncoupled Basis Sets Last time: matrices for J 2 , J + , J , J z , J x , J y in jm j basis for J = 0, 1/2, 1 Pauli spin 1/2 matrices arbitrary 2 × 2 When M is ρ visualization of fictitious vector in fictitious B-field When M is a term in H idea that arbitrary operator can be decomposed as sum of J i . TODAY: 1. H SO + H Zeeman as illustrative 2. Dimension of basis sets JLSM J and LM L SM S is same 3. matrix elements of H SO in both basis sets 4. matrix elements of H Zeeman in both basis sets 5. ladders and orthogonality for transformation between basis sets. Necessary to be able to evaluate matrix elements of H Zeeman in coupled basis. Why? Because coupled basis set does not explicitly give effects of L z or S z . MI =+ aa 01 r r σ types of operators e.g. magnetic moment ( is a known constant or a function of r) how to evaluate matrix elements (e.g. Stark Effect) e.g. Spin - Orbit J q JJ r 12
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25 - 2 5.73 Lecture #25 revised 4 November, 2002 1 22 0 . () Hs s s SO n Zeeman zz z z z r B =⋅ =− + ≡− + ξ ζ γω l ll l l l l h Suppose we have 2 kinds of angular momenta, which can be coupled to each other to form a total angular momentum. The components of L,S, and J each follow the standard angular momentum commutation rule, but These commutation rules specify that L and S act like vectors wrt J but as scalars wrt to each other. Coupled uncoupled vs. m representations. jsm sm js l * matrix elements of certain operators are more convenient in one basis set than the other * a unitary transformation between basis sets must exist * limiting cases for energy level patterns will give a factor of h anomalous g - value of e r r r J L S jm m sm j s l l l and s will each give a factor of h ζω n l and are in rad / s 0 rr LS J L L JS S ,, , ,. [] = 0 ij k ijk
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This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.

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notes25 - 5.73 Lecture #25 25 - 1 HSO + HZeeman Coupled vs....

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