《量子力学》第四章 量子力学的应用（4.5）角动量耦合

4-5角动量耦合 角动量的本质 ■对易关系

4-5角动量 耦合 ◼ 角动量的本质 ◼ 对易关系

角动量基本对易关系 [./z,/]=i koK j2,]=O

角动量基本对易关系 , ] 0 ˆ [ [ , ] ˆ 2 = =  i i j ijk k i j j j j i e j j  

换组算符 (++广),j ) 2

换组算符 ) ˆ ˆ ( 2 1 ˆ ), ˆ ˆ ( 2 1 ˆ ˆ ˆ ˆ ˆ _) ˆ ( 2 ˆ 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2 0 2 0 + − + − + − − + + − + + = + = − = + + = = − = = + j j i j j j j j j j j j j j j j j i j j j j i j x y z x y x y 

1=2o 2/1=主 2,1=O 1=O

, ˆ ] 0 ˆ [ , ˆ ] 0 ˆ [ [ ˆ , ˆ ] ˆ [ ˆ , ˆ ] 2 ˆ 0 2 2 0 0 = = =  =    + − j j j j j j j j j j    

CSCO 2 O 7 2|兄m〉=兄2元m Olam=mh am> 2

CSCO ?, ? ˆ ˆ , ˆ } ˆ { 0 2 2 0 2 = = = = m j m m m j m m m j j           

对固定的xC20) |ml|≤√元 △ max nn ( l.2 2 772 兄=j(j+1)

对固定的 ( 0) ( 1) , 1,....., ( 1), 1,2,... 2 1 max 1  = + = − − + = − =  =  =   j j m j j j j s s j m m m  

归一化 jj)j Jol jm)=mhl jim) =√(j+1)-m(m+1)jm+ j/m)=√(+1)-m(m-1)m-1) m)=f(+1)j

归一化 j j m j j j m j j m j j m m j m j j m j j m m j m j j m m j m j j j j j j 2 2 0 ( 1) ˆ ˆ ( 1) ( 1) 1 ˆ ( 1) ( 1) 1 ˆ , 1 ,.....,    = + = + − − − = + − + + = − − − +

矩阵元 (,)mn=(m1,m)=h√(+1)-m(n+1) mn+1 +√八(j+1)-n(n-1)n1 2 (/,)mn=〈/m/,|jm) 2zV/(j+1)-n(n+1、6m+1 2 √j(j+1)-n( J J JO+Ih-d

矩阵元 m n m n z m n z m n m n y m n y m n m n x m n x m n j j j j j m j j n n j j n n i j j n n i j j m j j n j j n n j j m j j n j j n n       2 2 1 1 1 1 ) ( 1) ˆ ( ˆ ) ˆ ( ( 1) ( 1) 2 ( 1) ( 1) 2 ˆ ) ˆ ( ( 1) ( 1) 2 ( 1) ( 1) 2 ˆ ) ˆ (        = + = = − + − − = = + − + + + − − = = + − + − + − +

=1/2 i O

        − =         − =         = = = 0 1 1 0 , 0 0 , 1 0 0 1 2 1/ 2  1  2  3  i i j j   

轨道角动量 j=l=int eger x|m)=Ymn(无/)

轨道角动量 ( / ) int x lm Y x x lm j l eger l m    = = =