Linear Combinations of Spherical Harmonics

SetOptions[SphericalPlot3D,PlotPoints -> 30,ColorFunctionScaling->False,ViewPoint->{2.6, 0, 1.4},Axes->False,Boxed->False,Mesh->True,ImageSize->Medium];
SetOptions[Manipulate,BaseStyle->{FontSize->20}];
SetOptions[Manipulator,BaseStyle->{FontSize->20},Appearance->"Labeled"];

In this worksheet, you will examine combinations of the spherical harmonics, ((Subscript[Y, l])^m)(θ,ϕ).

As a refresher, the probability densities,|((Subscript[Y, l])^m)(θ,ϕ)((|)^2), of the first few harmonics are shown below.

Y[l_,m_,θ_,ϕ_]:=SphericalHarmonicY[l,m,θ,ϕ];
Manipulate[
Ylmsq[θ_,ϕ_]=FullSimplify[ComplexExpand[Conjugate[Y[l,m,θ,ϕ]]*Y[l,m,θ,ϕ]]];
maxx=Max[Table[Abs[Ylmsq[θ,ϕ]],{θ,0,π,π/20},{ϕ,0,2π,π/20}]];
SpherePlot1=SphericalPlot3D[1,{θ,0,Pi},{ϕ,0,2*Pi},ColorFunction->Function[{x,y,z,θ,ϕ,r},Hue[0.67*(1-Ylmsq[θ,ϕ]/maxx)]]];
SpherePlot2=SphericalPlot3D[Ylmsq[θ,ϕ],{θ,0,Pi},{ϕ,0,2*Pi},PlotRange->All,ColorFunction->Function[{x,y,z,θ,ϕ,r},Hue[0.67*(1-Ylmsq[θ,ϕ]/maxx)]]];
SpherePlot3=SphericalPlot3D[1+Ylmsq[θ,ϕ],{θ,0,Pi},{ϕ,0,2*Pi},ColorFunction->Function[{x,y,z,θ,ϕ,r},Hue[0.67*(1-Ylmsq[θ,ϕ]/maxx)]]];
YlmsqLabel=StringForm["|((Subscript[Y, ``])^(``))(θ, ϕ)((|)^2)",l,m];
Grid[{{YlmsqLabel},{SpherePlot1,SpherePlot2,SpherePlot3}}],
{{l,0},0,2,1},{{m,0},-l,l,1},TrackedSymbols:>{l,m}]

The following are normalized linear combinations of ((Subscript[Y, l])^m)(θ,ϕ)’s.

Ylmsum1:=(1/Sqrt[2])(Y[1,1,θ,ϕ]+Y[1,-1,θ,ϕ])
Ylmsum2:=(1/Sqrt[2])(Y[1,1,θ,ϕ]-Y[1,-1,θ,ϕ])
Ylmsum3:=(1/Sqrt[2])(Y[1,1,θ,ϕ]+Exp[I Pi/3]Y[1,-1,θ,ϕ])
Ylmsum4:=(1/Sqrt[4])(Y[0,0,θ,ϕ]+Sqrt[3]Y[1,0,θ,ϕ])
Ylmsum5:=(1/Sqrt[2])(Y[2,2,θ,ϕ]+Exp[I Pi/3]Y[2,-2,θ,ϕ])
Ylmsum6:=(1/Sqrt[2])(Y[4,4,θ,ϕ]+Exp[I Pi/3]Y[4,-4,θ,ϕ])

1a) What does it mean to add functions of two dimensions (θ and ϕ)?

1b) Based on your knowledge of what individual spherical harmonics look like, what do you think the above combinations will look like?

Various representations of |Ylmsum#(θ,ϕ)((|)^2):

Manipulate[
ChosenCombination[θ_,ϕ_]:=ToExpression[ToString[StringForm["Ylmsum``",YlmComboSelector]]];
Ylmsumsq[θ_,ϕ_]=FullSimplify[ComplexExpand[Conjugate[ChosenCombination[θ,ϕ]]*ChosenCombination[θ,ϕ]]];
maxx=Max[Table[Abs[Ylmsumsq[θ,ϕ]],{θ,0,π,π/20},{ϕ,0,2π,π/20}]];
YlmsumLabel=StringForm["Ylmsum``(θ,ϕ)=",YlmComboSelector ][ChosenCombination[θ,ϕ]];
YlmsumsqLabel=StringForm["|Ylmsum``(θ,ϕ)((|)^2)=",YlmComboSelector][Ylmsumsq[θ,ϕ]];
SpherePlot1=SphericalPlot3D[1,{θ,0,Pi},{ϕ,0,2*Pi},ColorFunction->Function[{x,y,z,θ,ϕ,r},Hue[0.67(1-Ylmsumsq[θ,ϕ]/maxx)]]];
SpherePlot2=SphericalPlot3D[Ylmsumsq[θ,ϕ],{θ,0,Pi},{ϕ,0,2*Pi},PlotRange->All,ColorFunction->Function[{x,y,z,θ,ϕ,r},Hue[0.67(1-Ylmsumsq[θ,ϕ]/maxx)]]];
SpherePlot3=SphericalPlot3D[1+Ylmsumsq[θ,ϕ],{θ,0,Pi},{ϕ,0,2*Pi},ColorFunction->Function[{x,y,z,θ,ϕ,r},Hue[0.67(1-Ylmsumsq[θ,ϕ]/maxx)]]];
Grid[{{YlmsumLabel},{},{YlmsumsqLabel},{},{SpherePlot1,SpherePlot2,SpherePlot3}}],
{YlmComboSelector,1,6,1},TrackedSymbols:>{YlmComboSelector}]

2) How do the radial probability distribution plots compare to what you expected them to look like?

3a) What happens when you change the relative phase of the combination’s components?

3b) What happens if you add the same phase to all components in the combination?

by Corinne Manogue, Kerry Browne, Mike Vignal, and Ian Founds
Copyright 2010 Corinne Manogue