====== Recreate DensityMatrixPlot ======
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asked by [[mailto:dwt2004@gmail.com|David Tam]] (2022/03/27 19:10)
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In Mathematica, I define the basis according to the documentation for DensityMatrixPlot:

lbasisF = 
  With[{l = 3}, 
   Flatten[Table[{SphericalHarmonicY[l, m, \[Theta], \[Phi]], 
      SphericalHarmonicY[l, m, \[Theta], \[Phi]]}, {m, -l, l}]]];

Now I can construct a density matrix with some randomly chosen numbers, and make a plot:

dmF = CFDensityMatrix[3, 1, 
  With[{vv = {0.7, 0, 0.3, 0, 0, 0.3, 0, 0, 0.1, 0.5, 0.1, 0, 0.2, 0}}, vv/Sqrt@Total[vv^2]]]

DensityMatrixPlot[dmF]

However, the following superposition shows that directly plotting in the orbital basis doesn't make the same plot:

Show[
 dmplot,
 SphericalPlot3D[ Conjugate[lbasisF].dmF.lbasisF, {\[Theta], 0, Pi}, {\[Phi], 0,(*2 Pi*)Pi}
  , PlotRange -> All, AspectRatio -> Automatic, AxesLabel -> {"x", "y", "z"}]
 ]

What is the right way to find the angular function as function of theta and phi, and how does DensityMatrixPlot avoid this problem?

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~~DISCUSSION|Answers~~