asked by David Tam (2022/03/27 19:10)
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?