Spherical polar coordinates ar、aa0、op、a +( + Ox Ox op sine coso- cosecoso a sinΦ 0 r 00 rsine00 a 0 cos0sinφ a coso a sinsin 的 a a0 rsine dd sine 02 =Cos0. or a0 D2 1 1 (S11n7○ 1 o2 r2 ar Or r2 sine d0 a0' r2sin20062 Ze2 n-e (4πE)r The Schrodinger equation 1 1 a Ze2 (sin 十 ay8π2m(E+ )w=0 or r2sin 00 ae r2sin20862 h2 4π8' 2.1.2 The solution --separation of variables Substitute P(r,0,Φ)=R(r)⊙(O)Φ（φ）into the equation 1 )+7sin0 1 a 1 y8πm(E+ Or sn0c%+7sn900伊 Ze2 )w=0 a0 π8I and multiply with r2 R(r)Θ(0)Φ（φ） 1 1 (sin 1 )+ R(r）ar (0)sine a0 a0 ao0+8(E+4 ()sin20 02 h2 e2r2=0 1 0 2aR(r)、, 8m(E AnEo ze 1 R(r)or Or h2 er=o0)sino o0 (sin) 1 aΦ() + a01 Φ()sin20ap2Spherical polar coordinates () ( ) ( ) r x xr x xp θ φ θ ∂ ∂∂ ∂ ∂ ∂ ∂ =+ + ∂ ∂∂ ∂∂ ∂∂ cos cos sin sin cos rr rsin θ φ φ θ φ θ θ φ ∂ ∂ ∂ =+ − ∂ ∂∂ cos sin cos sin sin y rr rsin θ φ φ θ φ θ θ φ ∂ ∂ ∂∂ = + ⋅+ ⋅ ∂ ∂ ∂∂ θ θ θ ∂ ∂ − ⋅ ∂ ∂ = ∂ ∂ z r r sin cos 2 2 2 2 2 2 2 2 sin 1 (sin ) sin 1 ( ) 1 θ θ φ θ θ θ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ∇ = r r r r r r 2 0 (4 ) n e Ze V πε r − = − The Schrödinger equation 2 2 2 2 2 2 22 2 2 0 11 1 8 ( ) (sin ) ( ) 0 sin sin 4 me Ze r E rr r r r h r ψ ψψ π θ ψ θ θ θ θ φ πε ∂∂ ∂ ∂ ∂ + + + += ∂∂ ∂ ∂ ∂ 2.1.2 The solution --- separation of variables Substitute Ψ(r, θ, φ) =R(r)Θ(θ)Φ(φ) into the equation 2 2 2 2 2 2 22 2 2 0 11 1 8 ( ) (sin ) ( ) 0 sin sin 4 me Ze r E rr r r r h r ψ ψψ π θ ψ θ θ θ θ φ πε ∂∂ ∂ ∂ ∂ + + + += ∂∂ ∂ ∂ ∂ and multiply with ( ) ( ) ( ) 2 R r Θ θ Φ φ r ) 0 4 ( ( ) 8 ( )sin 1 ) ( ) (sin ( )sin 1 ) ( ) ( ( ) 1 2 0 2 2 2 2 2 2 2 + + = ∂ ∂ Φ Φ + ∂ ∂Θ ∂ ∂ Θ + ∂ ∂ ∂ ∂ r r ze E h m r R r r R r r e πε π φ φ θ φ θ θ θ θ θ θ ] ( ) ( )sin 1 ) ( ) (sin ( )sin 1 ) [ 4 ( 8 ) ( ) ( ( ) 1 2 2 2 2 0 2 2 2 2 φ φ θ φ θ θ θ πε θ θ θ π ∂ ∂ Φ Φ + ∂ ∂Θ ∂ ∂ Θ + + = − ∂ ∂ ∂ ∂ r r ze E h m r R r r R r r e