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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!

Thanksenter image description here

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

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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

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Or

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

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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] &

enter image description here

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] &

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

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