573pset09

573pset09 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY 5.73...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY 5.73 Quantum Mechanics I Fall , 2002 Professor Robert W. Field Problem Set #9 DUE : At the start of Lecture on Friday , November 22. Reading : Angular Momentum Handouts C-TDL , pages 999-1024 , 1027-1034 , 1035-1042 Spherical components of a vector operator V ± 1 = m 2 1/2 V x ± iV y [] V 0 = V z Scalar product of two vector operators V∑W = ( 1) µ V −µ W µ µ . Scalar product of two tensor operators T 0 (0) A 1 ,A 2 = ( µ T µ ( ω ) A 1 T ± µ ( ω ) A 2 µ . Problems : 1. CTDL , page 1086 , #2. 2. CTDL , page 1089 , #7. 3. CTDL , page 1089 , #8. 4. A. d orbitals are often labeled xy , xz , yz , z 2 , x 2 –y 2 . These labels are Cartesian tensor components. Find the linear combinations of binary products of x , y , and z that may be labeled as T (2) +2 and T (2) 0 . B. There is a powerful formula for constructing an operator of any desired T ( ) M spherical tensor character from products of components of other operators
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Chemistry 5.73 Page 2 Problem Set #9 T M ( ) A 1 ,A 2 [] = A µ 1 ,M −µ 1 ω 1 ω 2 T µ 1 ω 1 () A 1 T M 1 ω 2 A 2 µ 1 where A is a Wigner or Clebsch-Gordan coefficient , which is related to 3-j coefficients as follows: j 1 j 2 j 3 m 1 m 2 m 3 ≡− (m 1 + m 2 ) =− 1 j 1 j 2 m 3 2j 3 + 1 1/2 A M 1 M 2 M 3 j 1 j 2 j 3 . Use the T ( ) M [A 1 , A 2 ] formula to construct the spherical tensor T (3) +2 and T (3) 0 components of f orbitals by combining products of linear combinations of Cartesian labeled d and p orbitals. In other words , combine T (2) [x , y , z] with T (1) [x , y , z] to obtain T (3) M as a linear combination of products of 3 Cartesian components.
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573pset09 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY 5.73...

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