y1[l_,m_,θ_,ϕ_] := SphericalHarmonicY[l,m,θ,ϕ]
y12[l1_,m1_,l2_,m2_,θ_,ϕ_,T_] := (((Abs[(Sqrt[0.5]) y1[l1,m1,θ,ϕ] +(Sqrt[0.5]) y1[l2,m2,θ,ϕ] (E^(2 π I T))]))^2)

Animate[SphericalPlot3D[y12[1,0,0,0,θ,ϕ,T],{θ,0.,π},{ϕ,0.,2 π},Axes->False,PlotRange->{{-.3,.3},{-.3,.3},{-0.35,0.35}}],{T,-.25,.70,.05}]

Animate[SphericalPlot3D[y12[2,1,1,1,θ,ϕ,T],{θ,0.,π},{ϕ,0.,2 π},Axes->False,PlotRange->{{-.3,.3},{-.3,.3},{-0.35,0.35}}],{T,-.25,.70,.05}]

Animate[SphericalPlot3D[y12[2,1,1,0,θ,ϕ,T],{θ,0.,π},{ϕ,0.,2 π},Axes->False,PlotRange->{{-.3,.3},{-.3,.3},{-0.35,0.35}}],{T,-.25,.70,.05}]

nframes=20;
start = -0.25;
final = +0.75;
Δ = (final - start)/nframes;
Clear[p]

Do[Block[{$DisplayFunction=Identity},
p[i]=SphericalPlot3D[y12[1,0,0,0,θ,ϕ,(i-1) Δ +start],{θ,0.,π},{ϕ,0.,2 π},Axes->False,PlotRange->{{-.3,.3},{-.3,.3},{-0.30,0.30}},Boxed->False,ViewPoint->{1.,1.,0.}]],{i,1,nframes,1}]

Do[mean[i]=NIntegrate[Cos[θ]*y12[1,0,0,0,θ,ϕ,(i-1) Δ +start]*Sin[θ],{θ,0.,π},{ϕ,0.,2 π},AccuracyGoal->6],{i,1,nframes,1}]

meanList=Table[mean[i],{i,1,nframes}];

ListPlot[meanList]

Show[p[1]]

Show[{p[1],p[2]}]

Plot[y12[1,0,0,0,θ,0,start+Δ],{θ,0,π}]

ShowAnimation[{p[1],p[2],p[3],p[4],p[5],p[6],p[7],p[8]}]

Show[GraphicsArray[Partition[Map[p,Range[nframes]],4] ] ]

?*Graphics*