Download Quantum Mechanics: Conservation & Relationships of Angular Momentum - Prof. Lucien M. Crem and more Study notes Quantum Mechanics in PDF only on Docsity! Chapter 11 Total Angular Momentum With possibilities of both orbital L and spin S angular momentum in a system we must consider that total angular momentum J is conserved J = L + S . In a system in which the Hamiltonian commutes with J and J2 , [ H , JI ]=0 and [ H , J2 ]=0 , we need to establish how J is related to L and S. 1) Given ! L x ! L = i" ! L and ! S x ! S = i" ! S then ! J x ! J = i" ! J , ! J follows the angular momentum algebra. ! J x ! J = ! L + ! S( ) x ! L + ! S( ) = ! L x ! L i" ! L #$% + ! S x ! S i" ! S #$% + ! Sx ! L + ! Lx ! S cancel to zero # $& %& = i" ! J ! J x ! J = i" ! J QED quod erat demonstrantum 2) J2,L2!" # $ = 0 J 2commutes with L2 L2 +S2 + 2LiS , L2!" # $ = L 2,L2!" # $ =0 #$& %& + L 2,S2!" # $ =0 #$& %& + 2 S X L X ,L2!" # $ [L2,L X ]=0 #$& %& + 2 S Y L Y ,L2!" # $ [L2,L Y ]=0 #$& %& + 2 S Z L Z ,L2!" # $ [L2,L Z ]=0 #$& %& = 0 3) J2,S2!" # $ = 0 J 2commutes with S2 L2 +S2 + 2LiS , L2!" # $ = L 2,L2!" # $ =0 #$& %& + S 2,S2!" # $ =0 #$& %& + 2 L X S X ,S2!" # $ [S2,S X ]=0 #$& %& + 2 L Y S Y ,S2!" # $ [S2,L Y ]=0 #$& %& + 2 L Z S Z ,S2!" # $ [S2,L Z ]=0 #$& %& = 0 4) J2,J Z ! " # $ = 0 or J 2,J I ! " # $ = 0 I = 1,2,3 J 2commutes with J x J Y J Z L2 +S2 + 2LiS , L Z +S Z ! " # $ = L 2,L Z ! " # $ =0 #$& %& + L 2,S Z ! " # $ =0 #$& %& + S 2,L Z ! " # $ =0 #$& %& + S 2,S Z ! " # $ =0 #$& %& + 2S X L X ,L Z !" #$ =% i"L Y #$& %& + 2S Y L Y ,L Z !" #$ =+ i"L X #$& %& + 2S Z L z ,L Z !" #$ =0 #$& %& + 2L X S X ,S Z !" #$ =% i"S Y #$& %& + 2L Y S Y ,S Z !" #$ =+ i"S X #$& %& + 2L Z LS z ,S Z !" #$ =0 # $& %& = %2i"S X L Y + 2i"S Y L X % 2i"L X S Y + 2i"L Y S X = 0 J S L JZ ! S and ! L have constant projections on ! J ! Li ! J = L2 + ! Li ! S = L2 + 1 2 J2 ! L2 !S2( ) = 1 2 J2 + L2 !S2( ) = 1 2 j( j +1) + l(l +1) ! s(s +1)( )"2 = cons tan t ! Si ! J = S2 + ! Li ! S = S2 + 1 2 J2 ! L2 !S2( ) = 1 2 J2 + L2 !S2( ) = 1 2 j( j +1) ! l(l +1) + s(s +1)( )"2 = cons tan t Set of Commuting Operators for the System- Symmetries of the Hamiltonian The complete set of commuting operators is then H, J 2 , J Z , L 2 , S 2 . Each operator represents a constant of motion for the system, and each represented by a quantum number n, j, m j , l , s The most general wave function can be written ! n , j , m j ,l , s = n, j, m j , l , s = C m L ,m S L,S Clesbch"Gordan Coef! n R(r ) " l,m l Ylm #$( ) "n,ml ,ms mj =ml +ms % s,mS & S #$% H n, j, m j , l , s = En n, j, mj , l , s J2 n, j, m j , l , s = j( j +1)&2 n, j, m j , l , s J Z n, j, m j , l , s = m j & n, j, m j , l , s L2 n, j, m j , l , s = l(l +1)&2 n, j, m j , l , s S2 n, j, m j , l , s = s(s +1)&2 n, j, m j , l , s Addition of Angular Momentum ! L = ! L 1 + ! L 2 = | L 1 + ! L 2 | ........ | L 1 ! ! L 2 | ! S = ! S 1 + ! S 2 = |S 1 + S 2 | ........ |S 1 ! S 2 | J = L + S = | L + S | ........ | L ! S | m J = !J, !J +1 ......., +J = 2 j +1 states Classification of J States for the Hydrogen Atom J=L+1/2 L = 0 1 2 3 J = L +1/ 2 : 2s+1L J L = S, P, D, F J = 0 +1/ 2 J = 1/ 2 m j = ±1/ 2 2S 1/2 J = 1+1/ 2 J = 3 / 2 m j = ±3 / 2, ±1/ 2 2P 3/2, 1/2 J = 1/ 2 m j = ±1/ 2 J = 2 +1/ 2 J = 5 / 2 m j = ±5 / 2 = ±3 / 2, ±1/ 2 2D 5/2, 3/2, 1/2 J = 3 / 2 m j = ±3 / 2, ±1/ 2 2D 3/2, 1/2 J = 1/ 2 m j = ±1/ 2 2D 3/2, 1/2 e- P+ r
