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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 . M I = + a a 0 1 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 a a J q J J r 1 2
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25 - 2 5.73 Lecture #25 revised 4 November, 2002 1 2 2 0 . ( ) H s s H s s SO n Zeeman z z z z z r B = = − + ( ) ≡ − ( ) + ( ) ξ ζ γ ω 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. j sm sm j s * 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 anomalous g - value of e r r r J L S jm m sm j s and s will each give a factor of ζ ω n and are in rad / s 0 ( ) r r L S J L L J S S , , , , .
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