本征函数： 1/2 s-[- Pl(cose) (4.32) (+ImD! (Pauling Wilson,Introduction to Quantum Mechanics, McGraw-Hill,New York,1935) Associated Legendre functions )a--.1-0124到 w coSO
本征函数： (cos ) ( | |)! ( ) ( | |)! ( ) | | / , m l m Pl l m l l m S 1 2 2 2 1 + + − = (4.32) (Pauling & Wilson, Introduction to Quantum Mechanics, McGraw-Hill, New York, 1935) Associated Legendre functions ( ) ( ) , ! ( ) | | | | | | | |/ l l m l m m l m l w dw d w l P w 1 1 2 1 2 2 2 = − − + + l = 0, 1, 2, … (4.33) w = cos
Examples: P例=12m-0w2-0 P°(w)=w乃'(w)=3w1-w2)V2 (4.34) P'(w)=(1-w2)2P2(w)=3-3w2
Examples: 2 2 2 1 2 1 2 1 1 2 1 2 2 0 1 0 2 2 0 0 1 3 3 3 1 3 1 2 1 1 P w w P w w P w w P w w w P w P w w = − = − = = − = = − ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) / / (4.34)