Naïvely this is what happens and it obviously is not helpful!
In[7]:= Conjugate[SphericalHarmonicY[1, 1, θ, ϕ]]
Out[7]= -(1/2) E^(-I Conjugate[ϕ]) Sqrt[3/(2 π)] Conjugate[Sin[θ]]
So, I tried stating initially that $\theta$ and $\phi$ are reals but still that doesn't seem to have helped any bit,
In[8]:= θ ∈ Reals; ϕ ∈ Reals;
In[9]:= SphericalHarmonicY[1, 1, θ, ϕ]
Out[9]= -(1/2) E^(I ϕ) Sqrt[3/(2 π)] Sin[θ]
In[10]:= Conjugate[SphericalHarmonicY[1, 1, θ, ϕ]]
Out[10]= -(1/2) E^(-I Conjugate[ϕ]) Sqrt[3/(2 π)] Conjugate[Sin[θ]]
Kindly tell me how to do this? (I want to calculate sums like $\sum\limits_{m=-\ell}^{\ell}\left| Y_{l,m} (\theta,\phi)\right|^2$.)
Simplify[Conjugate[SphericalHarmonicY[1, 1, θ, ϕ]], θ ∈ Reals && ϕ ∈ Reals]
$\endgroup$$Assumptions
andAssumptions
. $\endgroup$