notes27 - 5.73 Lecture #27 27 - 1 Wigner-Eckart Theorem...

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27 - 1 5.73 Lecture #27 updated November 4, 2002 Wigner-Eckart Theorem CTDL, pages 999 - 1085, esp. 1048-1053 Last lecture on 1e Angular Part Next: 2 lectures on 1e radial part Many-e problems What do we know about 1 particle angular momentum? * easy vs. hard basis sets * limiting cases, correlation diagram * pert. theory – patterns at both limits plus distortion TODAY: 1. Define Scalar, Vector, Tensor Operators via Commutation Rules. Classification of an operator tells us how it transforms under coordinate rotation. 2. Statement of the Wigner-Eckart Theorem 3. Derive some matrix elements from Commutation Rules Scalar S J = M = 0, M independent Vector V J = 0,±1, M = 0, ±1, explicit M dependences of matrix elements These commutation rule derivations of matrix elements are tedious. There is a more direct but abstract derivation via rotation matrices. The goal here is to learn how to use 3-j coefficients. 1. , JM i i j ijk k kJ Basis set definition all matrix elements in JM basis set. JJ J [] =→ h ε 2 12 . JJ J JLS HH =+ −− Coupling of 2 angular momenta coupled uncoupled basis sets transformation via plus orthogonality. Also more general methods. + example SO Zeeman
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27 - 2 5.73 Lecture #27 updated November 4, 2002 ω Like components( µ ) scalar "constant" 0 J = 0 µ = 0 vector 3 components 1 J = 1 µ= ↔ +↔− + () ↔+ 0 12 z xi y y / / tensor (2 ω + 1) components 2 n d 2 2 +2, …, –2 [ ω is "rank"] 3 r d 3 3 Classification of Operators via Commutation Rules with CLASSIFYING ANGULAR MOMENTUM Spherical Tensor Components [CTDL, page 1089 #8] … Definition: Example: J = L + S common sense ? (vector analysis) This classification is useful for matrix elements of in basis set. T µ ω JM JT T T ±µ µ± µµ [] =+ µ µ ± ,( ) ( ) , / ωω h h 11 1 z . , . . 10 4 rr LS L S J J J =∴ × × & act as scalar operators with respect to each other. 2. and act as vectors 3. acts as scalar gives components of a vector [Because is composed of products of components of two vectors, it could act as a second rank tensor. But it does not !] wrt wrt wrt [Nonlecture: given 1 and 2, prove 3] Once operators are classified (classifications of same operator are different wrt different angular momenta), Wigner-Eckart Theorem provides angular factor of all matrix elements in any basis set! specifies everything else redundant- usually omitted ′′ ′ = ′′ µ µ µ N J M NJM A N J NJ MM JJ TT , ω δ vector coupling coefficient reduced matrix element no , , no 123 4 43 44
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27 - 3 5.73 Lecture #27 updated November 4, 2002 TL L JL L L L L L T JT T + −− + ++ [] =− + −+ + =+ () [] 1 11 2 12 1 1 1 1 1 1 2 22 2 1 () / // / , , L xy zx y y x x y z i ii i i L LL hh h h JS SS S J JLS ,, = is vector wrt * triangle rule
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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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notes27 - 5.73 Lecture #27 27 - 1 Wigner-Eckart Theorem...

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