# Relation beteen harmonic polynomials and spherical harmonics

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?

• For the first question, see en.wikipedia.org/wiki/…. Commented Dec 4, 2023 at 16:35
• For the second question, the set of spherical harmonics is an orthonormal basis for the space of regular spherical functions (i.e., well-behaved (finite?) functions defined on $0\leq \theta \leq \pi$, $0\leq\phi\leq 2\pi$. Commented Dec 4, 2023 at 16:44
• What pushes me to search such decomposition to write the angular part $(x^{3}+y^{3})$ in terms of the spherical harmonics? You said you wanted to know the possible values of $L^{2}$ and $L_{z}$ and their probabilities. That is literally the same problem as writing the angular part of the function in spherical harmonics (if you then take the magnitudes squared).
– Buzz
Commented Dec 5, 2023 at 3:27
• I know that expressing the wave function as linear combination of spherical harmonics their coefficients correspond to the amplitudes but I mean: Is the only way to do so, or it's possible to express the spherical harmonics combination without using the harmonic polynomials decomposition? Commented Dec 5, 2023 at 10:09