Representando as funções de onda angulares

Fórmulas analíticas

SphericalHarmonicY[3,2,θ,ϕ]

SphericalHarmonicY[3,2,θ,ϕ] //ExpToTrig

LegendreP[3,3,x]

Gráficos 3D

Manipulate[SphericalPlot3D[Evaluate[SphericalHarmonicY[l,m,θ,ϕ] SphericalHarmonicY[l,-m,θ,ϕ]],{θ,0,Pi},{ϕ,0,2Pi},PerformanceGoal->"Quality",AspectRatio->Automatic],{l,0,7,1},{m,0,l,1}]

Manipulate[SphericalPlot3D[Evaluate[LegendreP[l,m,Cos[θ]] ^2],{θ,0,Pi},{ϕ,0,2Pi},PerformanceGoal->"Quality",AspectRatio->Automatic],{l,0,7,1},{m,0,l,1}]

Manipulate[SphericalPlot3D[Evaluate[Re[SphericalHarmonicY[l,m,θ,ϕ] ]],{θ,0,Pi},{ϕ,0,2Pi},PerformanceGoal->"Quality",AspectRatio->Automatic],{l,0,7,1},{m,0,l,1}]

Gráfico da parte θ do harmônico esférico

Manipulate[ Plot[LegendreP[l,m,Cos[θ]],{θ,0,Pi}],{l,0,7,1},{m,0,l,1}]

Funções de onda de verdade do átomo

p3meios[θ_,ϕ_]:=ClebschGordan[{1,1},{1/2,1/2},{3/2,3/2}]^2SphericalHarmonicY[1,1,θ,ϕ]SphericalHarmonicY[1,-1,θ,ϕ]

pmeio[θ_,ϕ_]:=ClebschGordan[{1,0},{1/2,1/2},{3/2,1/2}]^2SphericalHarmonicY[1,0,θ,ϕ]^2+ClebschGordan[{1,1},{1/2,-1/2},{3/2,1/2}]^2SphericalHarmonicY[1,1,θ,ϕ]SphericalHarmonicY[1,-1,θ,ϕ]

GraphicsRow[{SphericalPlot3D[Evaluate[p3meios[θ,ϕ] ],{θ,0,Pi},{ϕ,0,2Pi},PerformanceGoal->"Quality",AspectRatio->Automatic],SphericalPlot3D[Evaluate[pmeio[θ,ϕ] ],{θ,0,Pi},{ϕ,0,2Pi},PerformanceGoal->"Quality",AspectRatio->Automatic]}]