Plotting Spherical Harmonics

Often we see "shapes" of the spherical harmonics shown graphically in textbooks.  One way to do so is by plotting the probability density \(TraditionalForm\`\(\(|\)\(\(Y\_l\^m\)(θ, ϕ)\)\( | \^2\)\)\)as the distance from the origin for each angle (θ, ϕ).  This can be done quite easily with Mathematica, using ParametricPlot3D command.

l = 0

SphericalHarmonicY[0, 0, θ, ϕ]

ParametricPlot3D[(1/(4  π)) {Sin[θ] Cos[ϕ], 
          Sin[θ] Sin[ϕ], Cos[θ]}, {θ, 
        0, π}, {ϕ, 0, 2  π}, PlotPoints -> 50]

l = 1

SphericalHarmonicY[1, 0, θ, ϕ]

ParametricPlot3D[(3/(4  π)) Cos[θ]^2 {Sin[θ] Cos[ϕ], Sin[θ] Sin[ϕ], 
          Cos[θ]}, {θ, 0, π}, {ϕ, 0, 2  π}, 
      PlotPoints -> 50]

SphericalHarmonicY[1, 1, θ, ϕ]

ParametricPlot3D[(3/(8  π)) Sin[θ]^2 {Sin[θ] Cos[ϕ], Sin[θ] Sin[ϕ], 
          Cos[θ]}, {θ, 0, π}, {ϕ, 0, 2  π}, 
      PlotPoints -> 50]

l = 2

SphericalHarmonicY[2, 0, θ, ϕ]

ParametricPlot3D[(5/(16  π)) (((3  Cos[θ]^2 - 
                1))^2) {Sin[θ] Cos[ϕ], 
          Sin[θ] Sin[ϕ], Cos[θ]}, {θ, 
        0, π}, {ϕ, 0, 2  π}, PlotPoints -> 50]

SphericalHarmonicY[2, 1, θ, ϕ]

ParametricPlot3D[(15/(8  π)) (((Sin[θ] 
                Cos[θ]))^2) {Sin[θ] Cos[ϕ], 
          Sin[θ] Sin[ϕ], Cos[θ]}, {θ, 
        0, π}, {ϕ, 0, 2  π}, PlotPoints -> 50]

SphericalHarmonicY[2, 2, θ, ϕ]

ParametricPlot3D[(15/(32  π)) (((Sin[θ]))^4) {Sin[θ] Cos[ϕ], Sin[θ] Sin[ϕ], Cos[θ]}, {θ, 
        0, π}, {ϕ, 0, 2  π}, PlotPoints -> 50]