Considering a spinless particle moving in the 3D space and described by the wavefunction $$\psi(\vec{x})=(x^3+y^3)\frac{e^\frac{-r}{2\lambda}}{r^2}$$ where $\lambda$ is a lengthscale.
I want to know the possible values of the angular momentum squared $L^2$ on this state and the possible values of the projections $L_z$ on the z-axis. Thus, I want to determine the values of the probabilities $P_{(l,m)}$ for $L^2$ and $L_z$.
Introducing $x_{\pm}=x\pm iy$ and $x_3=z$ it's possible to write $$x^3+y^3=\frac{1}{8}[(x_++x_-)^3+i(x_+-x_-)^3]=$$$$=\frac{1}{8}[(1+i)x_+^3+(1-i)x_-^3+3(1-i)x_+^2x_-+3(1+i)x_+x_-^2]$$
Then, i search the two constants $a$ and $b$ such that $$x_+^2x_=ah_{3,1}+br^2x_+$$ $$x_+x_-^2=ah_{3,-1}+br^2x_-$$ Where $h_{3,1}$ and $h_{3,-1}$ are harmonic polynomials and the result, having noticed that $r^2=x_3^2+x_+x_-$, has to be $$a=\frac{1}{5}$$ $$b=\frac{4}{5}$$ Now is possible to write $$x^3+y^3=\frac{1}{8}[(1+i)x_+^3+(1-i)x_-^3+\frac{3}{5}(1-i)h_{3,1}+\frac{3}{5}(1+i)h_{3,-1}+\frac{12}{5}(1-i)x_+r^2+\frac{12}{5}(1+i)x_-r^2]$$
And because $$Y_{3,\pm3}=\mp\frac{1}{8}\sqrt{\frac{35}{\pi}}{\frac{x_\pm^3}{r^3}}$$ $$Y_{1,\pm1}=\mp\frac{1}{2}\sqrt{\frac{3}{2\pi}}{\frac{x_\pm}{r}}$$ $$Y_{1,\pm1}=\mp\frac{1}{8}\sqrt{\frac{21}{\pi}}{\frac{h_{3,\pm1}}{r^3}}$$ It's possible to rewrite $(x^3+y^3)$ making the dependence on the spherical harmonics eplicit in order to read easily the linear combination of the angular momentum eigenfunctions $$x^3+y^3=r^3[-\sqrt{\frac{\pi}{35}}(1+i)Y_{3,3}+\sqrt{\frac{\pi}{35}}(1-i)Y_{3,-3}+\frac{3}{5}(1-i)\sqrt{\frac{\pi}{21}}Y_{3,1}-\frac{3}{5}(1+i)\sqrt{\frac{\pi}{21}}Y_{3,-1}-(1-i)\frac{3}{5}\sqrt{\frac{2\pi}{3}}Y_{1,1}+(1+i)\frac{3}{5}\sqrt{\frac{2\pi}{3}}Y_{1,-1}]=r^3\Theta(\theta,\phi)$$
My questions are:
What is the relation between the spherical harmonics and the harmonic polynomials? I tried to search but i've not found anything satisfying. I just found this Harmonic polynomial
What pushes me to search such decompisition using the harmonic polynomials to write the angular part $(x^3+y^3)$ as linear combination of the spherical harmonics? Is the only way to do so, or it's possible without using the harmonic polynomials decomposition?
