# Lowering operator for spherical harmonics [duplicate]

I need to be able to generate all of the $$l=2$$ spherical harmonics using the lowering operator. The specific question is listed below. Any assistance would be much appreciated!

## marked as duplicate by Jens, Thies Heidecke, Community♦Dec 13 '18 at 17:55

• And how is this to the Wolfram Mathematica, the software to which this Q&A site is dedicated? – Henrik Schumacher Dec 12 '18 at 23:18
• You might have better luck on math.stackexchange or maybe physics.stackexchange. Unless you specifically want to do this using Mathematica. – b3m2a1 Dec 12 '18 at 23:36
• Is it homework? – Αλέξανδρος Ζεγγ Dec 13 '18 at 4:50
• Belongs to physics.stackexchange.com – Thies Heidecke Dec 13 '18 at 13:33

Y[l_, m_] :=
Y[l, m] = Nest[Simplify[Exp[-I*ϕ] (-D[#, θ] +
I*Cot[θ]*D[#, ϕ])] &, Sin[θ]^l*Exp[I*l*ϕ], l - m]

(table1 = Table[Y[l, m], {l, 0, 2}, {m, 0, l}]) // Grid[#, Frame -> All] &


Or

(table2 = table1 // TrigReduce) // Grid[#, Frame -> All] &


EDIT: As suggested by Alex Trounev, compare with the built-in SphericalHarmonicY

Table[SphericalHarmonicY[l, m, θ, ϕ], {l, 0, 2}, {m, 0, l}] //
Grid[#, Frame -> All] &


EDIT 2: For clarity, look at the ratio of the two functions

Table[SphericalHarmonicY[l, m, θ, ϕ]/Y[l, m], {l, 0, 5}, {m, 0, l}] //
Simplify // Grid[#, Frame -> All] &


• Bob can you add for comparison Table[SphericalHarmonicY[l, m, \[Theta], \[Phi]], {l, 0, 2}, {m, 0, l}] // Grid[#, Frame -> All] & – Alex Trounev Dec 13 '18 at 11:55