a.Φ（φ）equation dPD(0+mΦ(p)=0 dd2 Its complex form: Φ=Aetilmlo,letm=tml,Φ=Aeim Normalization d-e-mem=1 1 A 1V2π Φm eimp三 cosmo+- -cosmo 2元 √2π 2 Φm(p)=Φm(中+2r） The values of m must be eimdeim(2)eimd.eim2 m=0,±1，±2，… eim2x =1 m:magnetic quantum cos2mπ+isin2mn=1 number
a. () equation Its complex form: =Aei|m| , let m= |m|, =Aeim ( ) 0 ( ) 2 2 2 m d d m i e m A d A e e d Normalization im m im im m m cos 2 cos 2 1 2 1 2 1 1 2 2 0 2 0 * cos 2 sin 2 1 1 ( ) ( 2 ) 2 ( 2 ) 2 m i m e e e e e im im im im im m m The values of m must be m 0,1,2, m : magnetic quantum number
complex function Φm=Aemo (m=0,±1，+2，…)》 =Aem=Acosmo+iAsin mo Dm=Ae Imo=Acosmo-iAsinmo its linear combination ④1=Φm+Φm=2Ac0Smp=Bcosmo Φ2=Φm-Φ_m=2 iAsin mo=B'sinmo normalization condition Jo Φi0=(Bcosmp')d地=B'z=l B= π real function Φ1 cosmo sinmo
m m B d B m d B normalization condition iA m B m A m B m its linear combination Ae A m iA m Ae A m iA m Ae m m m m m im m im m im m sin 1 cos 1 1 ( cos ) 1 2 sin 'sin 2 cos cos cos sin cos sin ( 0, 1, 2, ) 2 1 2 2 2 0 1 * 1 2 0 2 1 real function complex function
sin 2a 2sina cosa sin2 a+cos2a =1 cos 2a 2cos2 a-1 cos(a±B)=cosa cos B干sina sin B sina+sin B=2sin cosa-B 2 sina-sin B=2cos -sin Q-B 2 2 cosa+cosB=2cos+B a-B 2 2 cosa-cos B=-2sin a+B a-B sin 2 2
cos( ) cos cos sin sin sin 2 2sin cos 2 cos 2 sin sin 2sin 2 sin 2 sin sin 2cos 2 cos 2 cos cos 2cos 2 sin 2 cos cos 2sin sin cos 1 2 2 cos 2 2cos 1 2
Table The solution ofΦ（φ）equation m complex form real form 1 1 0 Φ0= V2π Φ0= √2π 1 1 1 Φ1= Φ- cos √2π Vπ 1 -1 Φ1= e io √2π Φ= 1 π 2 ①2= 1 Φ= 1 √2元 Vπ 1 -2 Φ-2= Φ盟= 1 sin 20 √2π Vπ
Table The solution of () equation sin 2 1 2 1 2 cos 2 1 2 1 2 sin 1 2 1 1 cos 1 2 1 1 2 1 2 1 0 sin 2 2 2 cos 2 2 2 sin 1 1 cos 1 1 0 0 i i i i e e e e m complex form real form
b.(0)equation sine o (sin) +Bsin20=m2 ⊙(0)60 a0 When B=/(/+1),1=0,1,2,3,......(0)is a well behaved function, 80=c1-ry月a网 d,4阿x+1y] 1:angular momentum quantum number x=cos0 1 (21+1)(1-m) C= 2'11 2(I+m) Necessary condition: 1≥m necessary condition:1>|ml hence,1=0,1,2,3,...(s,p,d.fg,h...) m=0,1,0,1,←2-1,0,1,2…
b. () equation When =l(l+1), l =0,1,2,3,…… () is a well behaved function, l m l m l l m l c x x d d c x l l l m x m l m Necessary condition : ( ) ( ) 2 (2 1) 2 ! 1 cos ( ) (1 ) [( 1) ] 2 2 2 l : angular momentum quantum number necessary condition: l |m| hence, l = 0, 1, 2, 3,…(s,p,d,f,g,h…) m=0,(-1,0,1),(-2,-1,0,1,2)…… 2 2 ) sin ( ) (sin ( ) sin m