故本征值问题(1)的本征值为:+1,1=0,2, 本征函数为: 由(5)→y(x)=P0(x)=(1-x) d P(x),m=0.,1 d x 7(5) P()=p() P(x)=(-+2()=(1-x2)2,oP(s)=sme (x)=x d x 822 1()=0-x)22()=3-x) or p'lcos0)==sin 20 2(x)=3(1-x),or2(cs)=3si2故本征值问题(1)的本征值为：l(l+1),l=0,1,2,… 本征函数为： ( ) ( ) ( ) ( ) ( ) ( ) 2 2 1- , 0,1,... 5 m m m l l m d y x P x x P x m l dx 由(5) ® = = = ( ) ( ) 0 Pl l x = P x ( ) ( ) ( ) ( ) 1 1 1 2 2 2 2 1 1 1 (1 ) (1 ) , cos sin P x x d P x x P x x orP dx q q = ¢ ¢ = - = - = ( ) ( ) ( ) ( ) 1 2 3 -1 2 1 1 2 2 2 2 2 2 (1 ) 3 1- , x d P x x P x x dx ¢ = - = ( ) ( ) 2 3 cos sin 2 6 2 or P¢ q q = ( ) ( ) ( ) 2 2 2 2 2 2 P x = = 3 1- x ,orP cosq q 3sin