asin 000(0)b.@(0)equation+βsin?θ= msin ea00(0) 00The process to solve this equation is too complicated. However,thesolutionalwayshastherealform@(0) = NPlml(cos 0)7dl+m(21 +1) (1 -ml)!(1-z2)，Pll(z)=(N=d2(22-1)2'1!2(l +m)!Demanding β =l(I+1) with I =0,1,2,3,... to make (0) a well-behavedfunctionl：angularmomentumquantumnumber（角动量量子数）(m= 0, ±1, ...)necessarycondition:I≥[m]hence, 1= 0, 1, 2, 3,...(s,p,d,f,g,h...)12m =0, (-1,0,1), (-2,-1,0,1,2),
b. (q) equation The process to solve this equation is too complicated. However, the solution always has the real form, l : angular momentum quantum number (角动量量子数) necessary condition: l |m| (m= 0, 1，.) 2 2 ) sin ( ) (sin ( ) sin m q q q q q q q Demanding = l(l+1) with l =0,1,2,3,. to make (q) a wellbehaved function. (q ) (cosq ) m NPl l l m l m l m m l z dz d l z P z l m l l m N ( ) ! ( ) , ( ) ( )! ( ) ( )! / 1 2 1 2 2 1 2 2 2 1 2 hence, l = 0, 1, 2, 3,.(s,p,d,f,g,h.) m =0, (-1,0,1), (-2,-1,0,1,2),
Examples of @(0):Examples of @(0):①(0)①(0)11mmV10[1(3cos2 0-1)000274V215V6sin 0cos 0±110cosA2215V3O0±2±1sinsin 04212
Examples of (q): l m 2 0 ±1 ±2 Examples of (q): l m 0 0 1 0 ±1 (q) 2 1 cosq 2 6 sin q 2 3 sin q cosq 2 15(q) q 2 sin 4 15(3cos 1) 4 10 2 q
C.Solution of R eguationOa8元dRmeBHOrR(r) dr4元SolutionR(r) = N ·e-p/2Ll+i(p)(real!)n--[(n+ 1)! /?pk(-1)k+1L'l(p)=Z(n-l-1-k)!(2l+1+k)!k!k=0zo4元gh?&ao= 0.529 Ap = 2or,α =2=2m.enaoao ~ Bohr radius, and also the atomic unit (au) of lengthnecessarycondition:n≥l+1(I= 0, 1, 2, ...hence, n =l, 2, 3,.....n: Principal12quantumnumber
c. Solution of R equation 2 0 2 2 2 2 4 1 8 r r ze E h m r dR r r R r dr e ) ( ) ( ) ( ( ) Solution a0 ~ Bohr radius, and also the atomic unit (au) of length. o 2 2 0 0 0 0 529 A Z 4 2 & . m e a na r e ， necessary condition: n l +1 (l = 0, 1, 2, .) hence, n = 1, 2, 3,. n: Principal quantum number (real!) 1 0 2 2 1 1 1 2 1 1 n l k k l k n l n l k l k k n l L ( )!( )! ! [( )! ] ( ) ( ) ( ) ( ) / 2 2 1 l Ln l R r N e
ExamplesofR(r):2Zn=1,[-0-p/23/2Rio= 2(二)enaon=2, [=0p)e-p/2R2o2-21Finally, the Rydberg equation is obtained to define the energy,N2h?RE=R= 13.6eV228元m.a8元nnm.aoR is called Rydberg constant with the value of 13.6 eV12
R is called Rydberg constant with the value of 13.6 eV. n=1, l=0 Examples of R(r): n=2, l=0 eV π m a h R n Z R n Z m a h E e e 13.6 8 8 2 0 2 2 2 2 2 2 2 0 2 2 ( ) ， r na Z 0 2 3 2 2 0 1 0 2 / / ( ) e a Z R 3 2 2 0 2 0 2 2 2 1 / / ( ) ( ) e a Z R Finally, the Rydberg equation is obtained to define the energy
InLm(r, e, Φ) = R(r)@(0)@(Φ)Real HydrogenlikeWaveFunctions(1e-ZrlaIsn=1,/=0,m=071/212zr-Zr/2a2sn=2,/=0,m=04(2m）1/2a5/21(2)re-Zr/2acos02pzn=2,/=1,m=04(2）/2a5/21Zre-Zr/2a sinocosΦ2pxn=2,/=1,m=±14(2元）/2a15/21(Z)re-Zr/2a sinesin2py4(2元）/2a12Transformed from the complex form of @(@) !
n.l,m(r, q, f) R(r)(q)F(f) Table n=1,l=0,m=0 n=2,l=0,m=0 n=2,l=1,m=0 n=2,l=1,m=1 Transformed from the complex form of F(f) ！