Justification of a trick in solving PDEs arising in Physics

[PeterDonis]

I still do not see it entirely

A function that vanishes at infinity in three dimensions does not have to vanish at the same "rate" (dependence on $r$) in different directions as $r \rightarrow \infty$.

 

[fluidistic]

Thanks for all, PeterDonis. I think this is much clearer to me now. I would love to study some group theory, as it is used in Physics in a lot of areas, and my knowledge is lacking there.

 

[PeterDonis]

Thanks for all, PeterDonis. I think this is much clearer to me now.

You're welcome! Glad I could help.

 

[vanhees71]

Isn't this a bit too complicated? For one-particle wavefunctions, a (pure) state can only be spherically symmetric around the origin if it depends only on $r=|\vec{x}|$ which automatically makes to an eigenstate of $\hat{\vec{L}}^2$ to the eigenvalue $0$, i.e., $\ell=0$.

Any superposition of wave functions with $\ell=1$ (dipole or $p$ waves) is thus not spherically symmetric.