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(2) !"# $%&'()*+ ,-.(/ 0-.(/(123+ 045 & /'637 8%/0+09:*6' &(;:)<+37 %/ =6)<+37 &*4:<<+37> ?(18=%@34:1 '+%A=(&4B 0%A4: 9(-4 0+09:* 1 (-% ,-.(/ /'637 :;:-9=%<6'C 8%=,0&(123+37 045 ' 8%9:<31 (;: 0D:=+3&<4: 0+*:9=+3&<+* C -96=+37 '&(1 :*<: %//&4(.+'(<4: &(<4:/A,1 :*+> E6'3&(0 D,<-31 ( D(;%'( -()/:F% & :;:-9=%<6' /(1 : 045 =%&0:8(=%'(B <( 3&5@34 =(/4(;<2 4 -29%'2C 8=&+ 3&+* 3&5@34 -29%': &(;:):B A5/2 %/8%'4:/<4% %/ -296' θ C ϕ 4 θ C ϕ > G:09 %3&+'409: C ): %8:=(9%=+ %=A49(;<:F% *%*:<9, 85/, /;( 84:='0&:F% :1;:-91=%<,2 Lˆ (1)2 4 k /;( /=,F4:F% Lˆ (2) -%*,9,12 k. h i ˆ (1) , L ˆ (2) = 0, L m. F/+) 02 9% %8:=(9%=+ =6)<43&-%': /&4(.(123: <( =6)<: &*4:<<: > H8=6A,1*+ %/8%'4:/&4:B <( 8+9(<4:I 1 (- '+F;2/(12 09(<+ '.(0<: %8:=(9%=( 0,*(=+3&<:F% *%*:<9, 85/, Lˆ = k ˆ (1) + L ˆ (2) > G(- 045 %-(): C <(0&: =%&'()(<4( A 5/2 3(.-%'434: %F6;<: 4 8 %&%09(<2 0 .,0&<: L 9(-k ): /;(k -=596' ,.(*-%'+37> $%&8(9=&*+ &(9:* /'( %8:(=9%=+ Jˆ(1,2) /&4(.(123: ' J k =6)<+37 8=&:09=&:<4(37 k. 2 2 Jˆ(1) |j1 , m1 i = j1 (j1 + 1) |j1 , m1 i , Jˆ(2) |j2 , m2 i = j2 (j2 + 1) |j2 , m2 i , (1) (2) Jˆ3 |j1 , m1 i = m1 |j1 , m1 i , Jˆ3 |j2 , m2 i = m2 |j2 , m2 i ,. KJL>MN F/&4: /;( ,.(9'4:<4( 8=&+15;4@*+ ,-.(/ 1 :/<%09:-C ' -96=+37 ~ = 1 C ;,A O 1 (- -9% '%;4 O 8=&:0-(;%'(;4@*+ %8:=(9%=+ Jˆ → Jˆ /~ > H92/ =:;(31 : -%*,9(314 8=&+1*,12 8%09(B k k h i h i (1,2) (1) Jˆ , Jˆ(1,2) = i Jˆ(1,2) , Jˆ , Jˆ(2) = 0. KJL>JN k. m. kmn n. P8:=(9%= 3(.-%'49:F% *%*:<9, 85/, 1 :09 0,*2. k. m. KJL>QN F/&4: 8=&:& 1(1,2) %&<(3&+;4@*+ %8:=(9%= 1 :/<%09-%'+ /&4(.(123+ %/8%'4:/<4% ' 84:='0&:1 ;,A /=,F4:1 8=&:09=&:<4> R%<4):1 A5/&4:*+ %8,0&3&(B %8:=(9%=+ 1 :/<%09-%': C F/+) ,8=(0&S 3&( 9% &<(3&<4: <%9(315> ?(0&+* &(/(<4:* A5/&4: 0-%<09=,%'(<4: 09(<6' '.(0<+37 |j, mi %8:=(9%=( Jˆ2 4 Jˆ3 &: 09(<6' |j1 , m1i 4 |j2, m2 i > T 8%09(34 %8:=(9%=( KJL>QN '4/(B C ): 09(<(*4 '.(0<+*4 %8:=(9%=( Jˆ3 A5/2 09(<+ UVWXYZ[W\] |j1, m1 i |j2 , m2i > $&:3&+'4@34: (1) (2) Jˆk = Jˆk ◦ 1(2) + 1(1) ◦ Jˆk ,. (1) (2) Jˆ3 |j1 , m1 i |j2 , m2 i = Jˆ3 |j1 , m1 i ◦ 1(2) |j2 , m2 i + 1(1) |j1 , m1 i ◦ Jˆ3 |j2 , m2 i = (m1 + m2 ) |j1 , m1 i |j2 , m2 i .. KJL>^N R%'09(1 : 8+9(<4:I 3&+ 09(<+ 9: 02 09(<(*4 '.(0<+*4 Jˆ2 _ P-(&,1 : 045 ): <4: > `A+ 9% MLa.
(3) '+-(&(B &(840&*+ Jˆ2 1 (-% 1 1 Jˆ2 = Jˆ+ Jˆ− + Jˆ− Jˆ+ + Jˆ32 2 2 1 2 1 (1) (2) (1) (2) (1) (2) (1) (2) (1) (2) = J+ + J+ J− + J− + J− + J− J+ + J+ + J3 + J3 2 2 1 1 (1) (1) (1) (1) (2) (2) (2) (2) = J+ J− + J− J+ + J+ J− + J− J+ 2 2 2 (1) (2) (1) (2) (1) (2) + J+ J− + J− J+ + J3 + J3 , KJL>. N. F/&4: '+-%=&+09(;4@*+ D(-9 C ): J (1) 4 Jm(2) -%*,9,12> E+;43&*+ 9:=(& /&4(.(<4: 8%0&3&:F6;S <+37 0-.(/<4-6' 0,*+ KN <( 0k9(< 4;%3&+<%'+ |j , m i |j , m i I 1 1 2 2 . (1) (1) (1) (1) Jˆ+ Jˆ− + Jˆ− Jˆ+ |j1 , m1 i |j2 , m2 i = (j1 (j1 + 1) − m1 (m1 − 1)) |j1 , m1 i |j2 , m2 i. + (j1 (j1 + 1) − m1 (m1 + 1)) |j1 , m1 i |j2 , m2 i = 2 j1 (j1 + 1) − m21 |j1 , m1 i |j2 , m2 i KJL>N. 4 (<(;%F43&<4:. KJL>N. (2) (2) (2) (2) Jˆ+ Jˆ− + Jˆ− Jˆ+ |j1 , m1 i |j2 , m2 i = 2 j2 (j2 + 1) − m22 |j1 , m1 i |j2 , m2 i .. T -%;:4. (1) (2) Jˆ+ Jˆ− |j1 , m1 i |j2 , m2 i p p = j1 (j1 + 1) − m1 (m1 + 1) j2 (j2 + 1) − m2 (m2 − 1) |j1 , m1 + 1i |j2 , m2 − 1i , (1) (2) Jˆ− Jˆ+ |j1 , m1 i |j2 , m2 i p p = j1 (j1 + 1) − m1 (m1 − 1) j2 (j2 + 1) − m2 (m2 + 1) |j1 , m1 − 1i |j2 , m2 + 1i. 4 '=:0&34: . (1) (2) Jˆ3 + Jˆ3. 2. KJL>N KJL>aN KJL>MLN. |j1 , m1 i |j2 , m2 i = m21 + 2m1 m2 + m22 |j1 , m1 i |j2 , m2 i .. T(9:* ' 0,*4: %9=&+*,1 :*+. N. Jˆ2 |j1 , m1 i |j2 , m2 i KJL>MM = (j1 (j1 + 1) + j2 (j2 + 1) + 2m1 m2 ) |j1 , m1 i |j2 , m2 i p p + j1 (j1 + 1) − m1 (m1 + 1) j2 (j2 + 1) − m2 (m2 − 1) |j1 , m1 + 1i |j2 , m2 − 1i p p + j1 (j1 + 1) − m1 (m1 − 1) j2 (j2 + 1) − m2 (m2 + 1) |j1 , m1 − 1i |j2 , m2 + 1i .. E4/&4*+ &(9:* C ): ' %F6;<%@34 ' '+<4-, /&4(.(<4( %8:=(9%=( Jˆ2 <( 09(< 4;%3&+<%'+ / 09 1 :*+ <%': 09(<+C *4(<%'434: |j , m + 1i |j , m − 1i 4 |j , m − 1i |j , m i |j , m i % ( 1. 1. 2. 2. 1. MML. 1. 2. 2. 1. 1.
(4) > E =9. ) B ): 0 09 4: 9: 09 < /8 4 / 12 9:1 0 *:1. =9 4. ( % &(,'( + C ' &+ ( + % %' ( ( ( '( %@3 |j2 , m2 + 1i ; 4 B / 0< > < ) ) * 1 1 :F : : F 9 : : I '.( Jˆ3 m1 + m2 C 3& % ( (.% %3& - '( C + Jˆ3 -% , , & Jˆ2 T =6'<(<4( JL>MMN '+<4-(C ): 09(<+ /4(F%<(;4&,123: Jˆ2 A5/2 ' %F6;<%@34 -%*A4<(31 (*4 ;4<4%'+*4 09(<6K' 4;%3&+<%'+37 |j , m i |j , m i I 1 1 2 2 X |j, mi = |j1 , m1 i |j2 , m2 i Cjjm , KJL>MJN 1 m1 ,j2 m2 m1 ,m2 m1 +m2 =m. F/&4: 0,*( 8=&:A4:F( 8% m1 4 m2 & '(=,<-4:* C ): m1 + m2 = m > E086.3&+<<4-4 ;43&A%': *<%)23: 09(<+ |j , m i |j , m i ' 9+37 -%*A4<(31 (37 ;4<4%'+37 <%0&2 <(&'5 Cjjmm ,j m \ XYZ[[U \ V] X W1
(5) 1[> 2 ;(2 '086.3&+<<4-6' 9+37 8=&+159% 09%0%'(B <4:3% 4<<2 <%9(315
(6) j1 j2
(7)
(8) j jm JL>MQN Cj m ,j m = . K
(9) m1 m2 m E6'3&(0 '&6= KJL>MJN 8=&+1*,1 : 8%09(B I 1. 1. 2. 2. 1. |j, mi =. 1. 2. X. 2. |j1 , m1 i |j2 , m2 i. m1 ,m2 m1 +m2 =m.
(10) j1 j2
(11)
(12) j , m1 m2
(13) m. . KJL>M^N. T(,'()*+C ): 09(<+ KJL>MJN 3&+ KJL>M^N 02 9(-): 09(<(*4 '.(0<+*4 J (1,2) 4 3&(0(*4 02 %&<(3&(<: 1 (-% X. |j1 , j2 ; j, mi =. |j1 , m1 i |j2 , m2 i. m1 ,m2 m1 +m2 =m. 2.
(14) j1 j2
(15)
(16) j . m1 m2
(17) m. . KJL>M N. E /(;0&:1 3&5@34 /;( ,8=%0&3&:<4( A5/&4:*+ ,)+'(B KJL>MJNC KJL>M^N>. !"# `A+ '+;43&+B C jm &(09%0,1 :*+ 0+09:*(9+3&<2 8=%3:/,=5 1 ('<:1 -%<09=,-314 09(<6' j m ,j m '.(0<+37 %8:=(9%=( Jˆ2 > T(,'()*+ <(184:='C ): 409<4:12 /'( 09(<+ 4;%3&+<%': C -96=: 02 09(<(*4 '.(0<+*4 Jˆ2 C *4(<%'434: 09(<+ % <(1 '+)0&+37 4 <(1<4)0&+37 *%);4'+37 =&,9(37 -=59, m = ±j > $&:3&+'4@34: /;( 9+37 /', '(=9%@34 %9=&+*,1 :*+ 1. 1,2. 1. 2. 2. 1,2. Jˆ2 |j1 , ±j1 i |j2 , ±j2 i = (j1 (j1 + 1) + j2 (j2 + 1) + 2j1 j2 ) |j1 , ±j1 i |j2 , ±j2 i = (j1 + j2 )(j1 + j2 + 1) |j1 , ±j1 i |j2 , ±j2 i .. KJL>MN E4/&4*+ &(9:* C ): 9: /'( 09(<+ 02 09(<(*4 '.(0<+*4 %8:=(9%=( Jˆ2 /% '(=9%@34 '.(0<:1 = 8:= 9 = / =9 4 0<:1 m = ±(j + j ) > R%&%09(1 : &(9:* 1 :S j = j1 + j2 % (& % ( % ( Jˆ3 % '( %@3 '.( /+<4: ,09(;4B '086.3&+<<4- 8=%8%=31 %<(;<%@34 8%*45/ &+ 09(<:1* |j2 = j1 + j2 , m = j1 + j2 i ( 09(<:* |j1 , j1i |j2 , j2i > T: '&F;5/, <( '(=,<:- ,<%=*%'(<4( '086.3&+<<4- 9:< *%): 8=&+1*%'(B 1 :/+<4: 8%09(B D(&+ eiϕ > =(/+3+1<4: D(&5 95 8=&+1*,1 : 045 1 (-% &:=%C 3&+;4 |j = j + j , m = j + j i = |j , j i |j , j i . KJL>MN 1. 2. 1. 2. MMM. 1. 1. 2. 2.
(18) E(=,<:- 9:< 9.,*(3&+ 045 <( '(=9%@B 84:='0&:F% '086.3&+<<4-( ;:A037(S%=/(<( KJL>MN E086.3&+<<4- 8=%8%=31 %<(;<%@34 /;( 09(<, % m = −j −j '+1/&4: <(* ' 08%06A <(9,=(;<+ & -%<09=,-314 09(<6' '.(0<+37 %8:=(9%=( Jˆ2 > E&6= K1JL>M2N 1 :09 84:='0&2 & 9&' > -%<':<314 %</%<(SH7%=9;:+(> (123 09(< *%):*+ 0-%<09=,%'(B '0&+09-4: 09(<+ '.(0<: %8:=(9%=( ˆ2 /% '(=S 9%@34 '.(0<:1 j =|j, jji1 + j2 8%8=&:& '4:;%-=%9<: /&4(.(<4: %8:=(9%=:* Jˆ− > &4(.(J123 1 :/<%S -=%9<4: %9=&+*,1 :*+ .
(19) j1 j2
(20)
(21) j = j1 + j2 = 1. j1 j2
(22) m = j1 + j2. Jˆ− |j, ji =. p 2j |j, j − 1i , p p (1) (2) Jˆ− + Jˆ− |j1 , j1 i |j2 , j2 i = 2j1 |j1 , j1 − 1i |j2 , j2 i + 2j2 |j1 , j1 i |j2 , j2 − 1i ,. KJL>MaN F/&4: j = j1 + j2 > R%<4:'() %A( 09(<+ ': '&%=&: KJL>MaN 02 0%A4: =6'<: C %9=&+*,1 :*+C ): KJL>JLN (9'% 045 ,8:'<4B C ): 9(- &/:<4%'(<+ 09(< |j, j − 1i 1 :09 8%8=('<4: ,<%=*%'(<+> E&6= KJL>JLN 9.,*(3&+ 045 <( '(=9%@34 %'4:/<437 '086.3&+<4-6' ;:A037(S%=/(<(I |j, j − 1i =. s. j1 |j1 , j1 − 1i |j2 , j2 i + j. s. j2 |j1 , j1 i |j2 , j2 − 1i . j.
(23) s
(24) j1 j1 j2
(25) j1 + j2 = ,
(26) j1 − 1 j2 j1 + j2 − 1 j1 + j2
(27) s j2 j1 j2
(28)
(29) j1 + j2 . =
(30) j1 j2 − 1 j1 + j2 − 1 j1 + j2 . (123 /% /+08%&+314 /'( 09(<+ % 3(.-%'49+* m = j. 1. + j2 − 1. KJL>JMN I. KJL>JJN *%):*+ 0-%<09=,%'(B /=,F2 -%*A4<(315 ;4<4%'2C -96=( A5/&4: %=9%F%<(;<( /% KJL>JLN> %*A4<(31 ( 9( A5/&4: 09(<:* '.(0<+* %8:=(9%=( Jˆ /% 9:1 0(*:1 '(=9%@34 '.(0<:1 m = > R <4: ) 09 < <4: 02 &/:F:<:=%'(<: C *3,04 %<( %/8%'4(/(B 09(<%'4 % 4<<+* j − 1 % '( ( + |j, mi > G:/+<+* *%);4'+* 09(<:* 1 :09 9, 09(< |j − 1, j − 1i > ` &(9:* j |j − 1, j − 1i = α |j1 , j1 − 1i |j2 , j2 i + β |j1 , j1 i |j2 , j2 − 1i , KJL>JQN F/&4: Y]XYZ\U ] K9% -%;:1<( -%<':<31 (N '086.3&+<<4-4 α 4 β '+;43&+*+ & '(=,<-, %=9%S F%<(;<%@34 s s j1 j2 +β =0 hj − 1, j − 1 |j, j − 1i = α KJL>J^N |j1 , j1 − 1i |j2 , j2 i ,. |j1 , j1 i |j2 , j2 − 1i. j. MMJ. j.
(31) 4 ,<%=*%'(<4(. KJL>J N G(- .(9'% 045 8=&:-%<(B =%&'42&(<4: =6'<( KJL>J^N 4 KJL>J N <4: 1 :09 1 :/<%&<(3&<: hj − 1, j − 1 |j − 1, j − 1i = α2 + β 2 = 1.. KJL>JN ,04*+ &(9:* 8=&+12B -%;:1<2 -%<':<315 C -96=( 1 :/<%&<(3&<4: %-=:@;4 '+A6= &<(-, ' =6'<(<4, KJL>JN> `A+ 95 -%<':<315 '+F%/<4: '+8%'4:/&4:B C &(,'()*+ ): s. α=±. α=. . j2 , j. s. β=∓. j1 . j.
(32) j1 j2
(33)
(34) j1 + j2 − 1 j1 j2 , β=
(35) j1 − 1 j2 j1 + j2 − 1 j1 j2 − 1.
(36)
(37) j1 + j2 − 1
(38)
(39) j1 + j2 − 1 .. KJL>JN. R=&+1*4:*+C ): '086.3&+<<4- β > 0 C 3% %F6;<4: *%):*+ &(840(B 1 (-%. KJL>JN F/&4: 0 6 n 6 2j2 > %<':<31 ( 9( 8%&'(;( <(* '+&<(3&+B 1 ,) 1 :/<%&<(3&<4: '0&+09-4: '086.3&+<<4-4 ;:A037(S%=/(<(> H92/ *(*+ . s. |j − 1, j − 1i = −. 3&+;4.
(40) j1 j2
(41)
(42) j1 + j2 − n > 0, j1 j2 − n
(43) j1 + j2 − n. j1 |j1 , j1 − 1i |j2 , j2 i + j. s. j2 |j1 , j1 i |j2 , j2 − 1i , j. KJL>JaN. s
(44)
(45) j1 j1 j2
(46) j1 + j2 = − , j1 − 1 j2
(47) j1 + j2 − 1 j1 + j2
(48) s
(49) j2 j1 j2
(50) j1 + j2 . = j1 j2 − 1
(51) j1 + j2 − 1 j1 + j2. . KJL>QLN (;0&( 8=%3:/,=( 1 :09 1 ,) %3&+'409( I /&4(.(*+ <( 09(<+ JL>JLN 4 JL>JaN %8 :=(9%=:* K K 4 /%09(1 :*+ /'4: -%*A4<(31 : ;4<4%': 09(<6' Jˆ− |j1 , j1 − 2i |j2 , j2 i , |j1 , j1 − 1i |j2 , j2 − 1i , |j1 , j1 i |j2 , j2 − 2i KJL>QMN %/8%'4(/(123: 09(<%* |j, j − 2i ,. |j − 1, j − 2i ,. F/&4: j = j1 +j2 > %=&+09(123 & '(=,<-, %=9%F%<(;<%@34 C ,<%=*%'(<4( 4 -%<':<314 KJL>JN -%<09=,,1 :*+ 9=&:342 -%*A4<3(15 ;4<4%'2 %/8%'4(/(1232 09(<%'4 |j − 2, j − 2i .. MMQ.
(52) T(,'()*+C ): ' &:=%'+* -=%-, /+08%<%'(;4@*+ 1 :/<+* 09(<:* 4;%3&+<%'+* |j , j i 1 84:='0&+* /'%*( 09(<(*4 KJL>JJNC ( ' /=,F4* 9=&:*( 09(<(*4 KJL>QMN1> E |j2 , j2 i C ' 9 * = 8 4<<4@*+ *4:B /% /+08%&+314 n + 1 09(<6' I n− + - %-, %' |j1 , j1 i |j2 , j2 i |j1 , j1 i |j2 , j2 − 1i , |j1 , j1 i |j2 , j2 − 2i ,. |j1 , j1 − 1i |j2 , j2 i. |j1 , j1 − 1i |j2 , j2 − 1i ,. |j1 , j1 − 2i |j2 , j2 i. .... KJL>QJN. |j1 , j1 i |j2 , j2 − ni , |j1 , j1 − 1i |j2 , j2 − n + 1i , . . . |j1 , j1 − ni |j2 , j2 i. 4 9(- /(;:1> G:):;4 1 :/<(- n = 2j ;,A n = 2j 9% '6'3&(0 ' <(0958<+* Kn + 1O'0&+*N -=%-, <4: 8%1 ('4( 045 1 :/:< K;,A1 /'( 1 :@;4 j1 2= j2 N & /', 0-=(1<+37 09(<6' ': '&%=&: JNC 3% %&<(3&(C ): ;43&A( 09(<6' 1 :09 9(-( 0(*( K;,A % 1 :/:< *<4:10&(N 1 (- ' -=%-, K8J%L>Q 8=&:/<4*> ` 9% & -%;:4 %&<(3&(C ): <4: *,04*+ -%<09=,%'(B <%':1 -%*A4<(314 ;4<4%':1 C -96=( %/8%'4(/(.(A+ 09(<%'4 |j − n − 1, j − n − 1i > P/ 9:F% *%*:<9, 8=&:09(1 : 8%1 (S '4(B 045 -%<4:3&<%@B -%<09=,%'(<4( 09(<6' % <%'+* j > &+;4 C ): *4<4*(;<: jmin = j1 − j2 F/+ j1 > j2 C ;,A jmin = j2 − j1 C F/+ j2 > j1 > P/ 9:1 8%=+ ;43&A( 09(<6' %9=&+*+'(<( ' -%S ;:1<+37 -=%-(37 A5/&4: 045 &*<4:10&(B C () %042F<4:*+ %09(9<4 C 8%1 :/+<3&+ 09(< 4;%3&+<%'+ > |j1 , −j1 i |j2 , −j2 i R%/0,*,1*+ I ' '+<4-, %840(<:1 -%<09=,-314 %9=&+*(*+ -%*A4<(31 : ;4<4%': 09(<6' 4;%3&+<%'+37 %/8%'4(/(123: 09(<%* '.(0<+* 3(.-%'49:F% *%*:<9, 85/, % j &('(=9+* *45/&+ |j1 − j2 | 6 j 6 j1 + j2 . KJL>QQN 43&A( '0&+09-437 09(<6' 4;%3&+<%'+37 '+<%04 (2j1 + 1) (2j2 + 1) , KJL>Q^N <(9%*4(09 ;43&A( 09(<6' '.(0<+37 %8:=(9%=( 3(.-%'49:F% -=59, Jˆ2 4 Jˆ '+<%04 3. KJL>Q N. j1 +j2. N=. X. (2j + 1).. j=|j1 −j2 |. T(.6)*+C ): j > j 4 %A;43&*+ N &: '&%=, <( 0,*5 8%0958, (=+9*:9+3&<:F% I 1 2 j1 +j2. N =2. X. j1 +j2. j+. X. 1 = [(j1 + j2 ) + (j1 − j2 )] [2j2 + 1]+(2j2 + 1) = (2j1 + 1) (2j2 + 1) .. JL>QN K 43&A( 09(<6' 4;%3&+<%'+37 4 ;43&A( 09(<6' '.(0<+37 %8:=(9%=( 3(.-%'49:F% -=59, Jˆ2 4 Jˆ 02 0%A4: =6'<: > E /%/(9-, %A( &A4%=+ 09(<6' 02 %=9%<%=*(;<: C 092/ 9=(<0D%=*(31 ( KJL>M^3N /;( j & 8=&:/&4(., KJL>QQN 1 :09 D(-9+3&<4: W W[W V[ [ W X A(&+ 4;%3&+<%':1 /% A(&+ 09(<6' '.(0<+37 %8:=(9%=( 3(.-%'49:F% -=59, Jˆ2 4 Jˆ > j=j1 −j2. j=j1 −j2. 3. MM^.
(53) # ! # ! T %840(<:1 '+):1 -%<09=,-314 '+<4-(C ): '086.3&+<<4-4 ;:A037(S%=/(<( 02 =&:3&+'409: 4 08=&5):<4: 7:=*49%'0-4: 09(<,
(54) X j1 j2
(55)
(56) j |j, mi = |j , m i |j , m i KJL>QN 1. 8=&+1*,1 : 8%09(B. 1. 2. 2. m1 m2
(57) m. m1 ,m2. KJL>QN m ,m F/&4: /;( 86<4:10&:1 '+F%/+ &(*4:<4;4@*+ j O9+ 4 mO+ <( '4:;-%@34 8=4*%'(<: > %=&+09(S 123 & '(=,<-, %=9%F%<(;<%@34 09(<6' hj 0 , m0 | j, mi = δjj δmm KJL>QaN %9=&+*,1 :*+
(58)
(59) X j1 j2
(60) j j1 j2
(61) j 0
(62)
(63) = δjj δmm . KJL>^LN 0 hj 0 , m0 | =. X 0 1. hj10 , m01 | hj20 , m02 |. 0 2. .
(64) j10 j20
(65)
(66) j 0 , m01 m02
(67) m0. 0. m1 m2
(68) m. 0. 0. m1 m2
(69) m. 0. T'42&:- 9:< 8%&'(;( <(* <( &(840(<4: =:;(314 %/'=%9<:1 /% KJL>M^N m1 ,m2. KJL>^MN j,m 8=&+ 3&+* 0,*%'(<4: 8% j 8=&:A4:F( ' F=(<43(37 KJL>QQNC ( 0,*%'(<4: 8% m %F=(<43&( 045 /% m = m + m > 2 T '(=,<-,1 ,<%=* %'(<4( 09(<, KJL>^MN %9=&+*,1 :*+ /=,F2 =:;(315 %=9%F%<(;<%@34 I
(70)
(71) X j1 j2
(72) j j1 j2
(73) j 0
(74)
(75) =δ δ . KJL>^JN |j1 , m1 i |j2 , m2 i =. j,m. m01 m02
(76) m. X. |j, mi. .
(77) j1 j2
(78)
(79) j , m1 m2
(80) m. m1 m2
(81) m0. m01 m1 m02 m2. !
(82) " &4(.(123 %8 :=(9%=:* < =6 < <4: KJL>QN %9=&+*,1 :*+ I Jˆ+ ( ' (. p j(j + 1) − m(m + 1) |j, m + 1i X hp = j1 (j1 + 1) − m1 (m1 + 1) |j1 , m1 + 1i |j2 , m2 i m1 ,m2.
(83) i j j2
(84)
(85) j 1 + j2 (j2 + 1) − m2 (m2 + 1) |j1 , m1 i |j2 , m2 + 1i m1 m2
(86) m
(87) X p j1 j2
(88)
(89) j = j1 (j1 + 1) − m1 (m1 − 1) |j1 , m1 i |j2 , m2 i m1 − 1 m2
(90) m m1 ,m2
(91) p
(92) j j1 j2
(93) + j2 (j2 + 1) − m2 (m2 − 1) |j1 , m1 i |j2 , m2 i m1 m2 − 1
(94) m p. MM. KJL>^QN.
(95) T(09%0,1*+ 9:=(& '&6= KJL>QN (;: /;( 09(<, |j, m + 1i 4 8%=6'<(1*+ '086.3&+<<4-4 8=&+ 09(<(37 |j , m i |j , m i I 1 1 2 2
(96) j1 j2
(97)
(98) j j(j + 1) − m(m + 1) m1 m2
(99) m + 1 p j1 j2 = j1 (j1 + 1) − m1 (m1 − 1) m1 − 1 m2
(100) p
(101) j1 j2
(102) + j2 (j2 + 1) − m2 (m2 − 1) m1 m2 − 1
(103).
(104)
(105) j
(106)
(107) m j . m.
(108) j1 j2
(109)
(110) j m1 m2
(111) m − 1 p j1 j2 = j1 (j1 + 1) − m1 (m1 + 1) m1 + 1 m2
(112) p
(113) j1 j2
(114) + j2 (j2 + 1) − m2 (m2 + 1) m1 m2 + 1
(115).
(116)
(117) j
(118)
(119) m j . m. p. . KJL>^^N. R%0958,123 ' (<(;%F43&<+ 08%06A /;( %8:=(9%9( Jˆ %9=&+*(*+ /=,F4 &'42&:- =:-,=:<3+1<+ − p j(j + 1) − m(m − 1). . KJL>^ N. # # # !
(120) !# # # E086.3&+<<4-4 ;:A037(S%=/(<( 02 09(A:;(=+&%'(<: 4 ' 9:1 D%=*4: *%)<( 1 : &<(;:B ' ;49:=(9,=&: > ?( 8=&+-.(/ /;( j = j = 1 9(A:;( 9(-( 8=&+1*,1 : 8%09(B 1 2 M J M L J M J j j1 = 1 j2 = 1 J J M M L L L −1 −1 −2 m m1 m2 J M M M M L 1 1 2 2 M L 1 1 − 2 2 M −1 JL>^N 1 1 1 K 6 2 3 L L 2 0 − 13 3 M 1 1 −1 − 12 6 3 L −1 1 1 2 2 L 1 1 −1 −2 2 M −1 −1 F/&4: '(=9%@34 09%123: ' 9(A:;4 <(;:)+ '&42B & 84:='4(09-4:* C ( :':<9,(;<+ &<(- − 8%09(S '4B 8=&:/ 84:='4(09-4:*> ?4: '+8:.<4%<: *4:103( 9% &:=(> ?( 8=&+-.(/ 09(< |1, 1i /(1 : 045 =%&840(B 1 (-% r r 1 1 JL>^N |1, 1i = |1, 1i |1, 0i − |1, 0i |1, 1i . K E(=9% <( -%<4:3 &(&<(3&+B C ): 2' ;49:=(9,=&: 08%92+-( 045 9&' > '086.3&+<<4-4 2j 4 3j. MM.
(121) MM $+0,<:- MI.
(122)
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