Quantum Mechanics: Angular Momenta Addition and Clebsch-Gordon Coefficients
Explore the addition of angular momenta in quantum mechanics, focusing on the total spin of spin-1/2 particles and Clebsch-Gordon coefficients. Understand the conservation of total angular momentum and discover the explicit formula for Clebsch-Gordon coefficients, along with details for a simple case. Dive into the implications of particle exchange and the degeneracies associated with different eigenstates.
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PHY 741 Quantum Mechanics 12-12:50 AM MWF Olin 103 Plan for Lecture 21: Addition of angular momenta Chap. 15 1. Total spin due to two spin-1/2 particles 2. Clebsch-Gordon coefficients 10/20/2017 PHY 741 Fall 2017 -- Lecture 21 1
10/20/2017 PHY 741 Fall 2017 -- Lecture 21 2
In the absence of external magnetic fields, the internal magnetic dipoles cause spin interactions within each system, however, the total angular momentum of the system should be conserved. Clebsch-Gordon coefficients , , m m = , , JM j 1 2 j j m j m j m j m J M j 1 2 j 1 1 2 2 1 1 2 2 , 1 2 Finding the total angular momentum addition of angular momentum. 1 2 2 2 JM J = = = + J J j j + ( 1) J JM J JM M JM z = = + 2 2 j ( 1) j m j j j m 1 j 1 j 1 1 1 1 1 m m j m 1 1 1 1 1 1 z 10/20/2017 PHY 741 Fall 2017 -- Lecture 21 3
Addition of angular momentum Clebsch-Gordon coefficients , m m = , , , JM j 1 2 j j m j m j m j m J M j 1 2 j 1 1 2 2 1 1 2 2 , 1 2 10/20/2017 PHY 741 Fall 2017 -- Lecture 21 4
Explicit formula for Clebsch-Gordon coefficients: 10/20/2017 PHY 741 Fall 2017 -- Lecture 21 5
10/20/2017 PHY 741 Fall 2017 -- Lecture 21 6
10/20/2017 PHY 741 Fall 2017 -- Lecture 21 7
Details for a simple case: Clebsch-Gordon coefficients , m m Recall that : = + M m m = , , , JM j 1 2 j j m j m j m j m J M j 1 2 j 1 2 1 1 2 2 1 1 2 2 , 1 2 ( ) = + + 2 2 1 J JM = J M J M J M J j j 1 2 = 11; , 1 1 2 2 1 1 2 2 = 1 1 2 2 11; 2 10; J 1 1 2 2 j 1 1 2 2 ( ) + = + , , , j 1 1 2 2 1 1 2 2 1 2 1 2 1 2 1 1 2 2 1 1 2 2 1 2 1 2 1 2 ( ) = + 10; , , 1 1 2 2 1 2 1 2 1 1 2 2 1 1 2 2 1 2 1 2 1 1; = , 1 1 2 2 1 2 1 2 1 2 1 2 10/20/2017 PHY 741 Fall 2017 -- Lecture 21 8
Summary of results: 11; = , 1 1 2 2 1 1 2 2 1 1 1 2 2 J=1 degeneracy: 2J+1=3 ( ) = + 10; , , 1 1 2 2 1 2 1 2 1 1 2 2 1 1 2 2 1 2 1 2 2 1 1; = , 1 1 2 2 1 2 1 2 1 2 1 2 J=0 degeneracy: 2J+1=1 1 ( ) = 00; , , 1 1 2 2 1 2 1 2 1 1 2 2 1 1 2 2 1 2 1 2 2 Note that these eigenstates have different behaviors wrt to particle exchange: 1 ; even under particle exchange M 1 1 2 2 00; odd under particle exchange 1 1 2 2 10/20/2017 PHY 741 Fall 2017 -- Lecture 21 9
General accounting: Total number of states : + j j 1 2 ( ) + + = + (2 1)(2 1) 2 1 j j J 1 2 = J j j 1 2 Case of "addition" of spin and orbital angular momentu l j s j = = = m 1 2 1 2 + + + l M l l M l 1 2 1 2 ( ) ( ) ( ) + = + + ; ; ; l M l l M l M 1 2 1 2 1 2 1 1 2 2 1 2 1 2 1 2 + + 2 1 2 + 1 + + l M l l M l 1 2 1 2 ( ) ( ) ( ) = + + ; ; ; l M l l M l M 1 2 1 2 1 2 1 1 2 2 1 2 1 2 1 2 + + 2 1 2 1 10/20/2017 PHY 741 Fall 2017 -- Lecture 21 10
Spin-orbit interaction due to spin alignment in magnetic field generated by orbital motion ( ) Note that: = ; ; SO JM ls JM ls G r = S L H G r SO J =S+L J S + + 2 2 2 L S L 2 = S L ( ) ; ; H G r JM ls JM ls 2 ( ) 2 ( ) + + + = ( 1) ( 1) ( 1) j j s s l l 2 ( ) G r ( ) ( ) + + = ; ; l M ls H l M ls l 1 2 1 2 SO 2 2( ) 2 G r ( ) ( ) ( ) = l + ; ; 1 l M ls H l M l s 1 2 1 2 SO 10/20/2017 PHY 741 Fall 2017 -- Lecture 21 11
Coupling of orbital angular momenta of multiple electrons J = L +L 1 2 ( ) = + + , 1,...., J l l l l l l 1 2 1 2 1 2 l = = Example: J J J = 1 total of 9 orbital states l 1 2 = = 0 1 state 1 3 states 2 5 stat s e 10/20/2017 PHY 741 Fall 2017 -- Lecture 21 12
Consequences of orbital coupling on energies of multi- electron atoms Example C 1s2 2s2 2p2 https://physics.nist.gov/PhysRefData/ASD/levels_form.html 10/20/2017 PHY 741 Fall 2017 -- Lecture 21 13
+ Atomic term notation: (2 1) S J L L symbol spin -------------------------------------------- 0 S S=0 1 P S=1 2 D S=0 3 F S=1 10/20/2017 PHY 741 Fall 2017 -- Lecture 21 14
Example for C 1s2 2s2 2p2 l + + = States of total o rbital momen m: (2 tu 1)(2 1) 9 l 1 2 6 5/ 2 15 = Total number of orbital and spin configurations: 5 states 1 states 9 state s P 1 D 1 S 3 Energetic differences are due to the electron-electron Coulomb repulsion: e V r r + + 2 4 r = = * r r ( ) ( ) Y Y 1 2 ee 1 2 1 r 1 2 10/20/2017 PHY 741 Fall 2017 -- Lecture 21 15
Example for 1D state = = ; 22;11 , , ; JM l 1 2 l l m l m l m l m JM ll 1 1 2 2 1 1 2 2 1 2 , m m 1 2 = r r = 11 ,11 ( ) ( ) ( ) ( ) R r R r Y Y 2 1 2 2 11 1 11 2 p p 11,11 11,11 V ee 10/20/2017 PHY 741 Fall 2017 -- Lecture 21 16
