Common eigenkets of spherically symmetric Hamiltonian

Question

In a QM text it states: "Consider a spinless particle subjected to a spherical symmetrical potential. The wave equation is known to be separable coordinates, and the energy eigenfunctions can be written as $$\langle \mathbf{x'}| n,l,m \rangle = R_{nl}(r)Y_{l}^{m}(\theta, \phi)$$ where the position vector $\mathbf{x}'$ is specified by the spherical coordinates $r, \theta$ and $\phi$."

Can we consider that we obtain this by the following steps: $$|n,l,m \rangle = |n,l \rangle \otimes |l,m \rangle$$ hence $$\langle x'|n, l, m \rangle = (\langle \vec{r}| \otimes \langle \theta, \phi|)(|n, l \rangle \otimes |l, m \rangle) = (\langle \vec{r}| n, l \rangle)\otimes(\langle \theta, \phi | l, m \rangle) = R_{nl}(r)Y_{l}^{m}(\theta, \phi)$$

Is what I wrote nonsense or is there something there? The question was motivated by the fact that $|l,m\rangle = Y_{l}^{m}(\theta, \phi)$...hence what we have left is $R_{nl}(r)$.

Thanks.

Answer 1

Yeah, that's a reasonable way to understand that structure. The reason you don't see it that often is because it's not very useful, but it's a valid analysis.