In the expansion of the scattered wave function, why do these two functions have the same index?

Question

See Griffiths Quantum Mechanics, eq. 11.21. Evidently, $$\psi(r,\theta,\phi)=Ae^{ikz}+A\sum\limits_{l,m}^{\infty}C_{l,m}h_{l}(kr)Y_{l}^{m}(\theta,\phi).$$ But I don't see why the $l$th Hankel function necessarily needs to multiply the $l$th spherical harmonic. Seeing as we arrived at this equation through separation of variables, my feeling is that the equation should look something like: $$\psi(r,\theta,\phi)=Ae^{ikz}+A\sum\limits_{l,m,n}^{\infty}C_{l,m,n}h_{n}(kr)Y_{l}^{m}(\theta,\phi).$$

Answer 1

The indices must match because even though assuming a product solution ultimately gives independent equations for each of the factors, the overall product must satisfy the initial differential equation, and this will only happen if the indices match. To see how this arises in the radial equation, notice that the radial equation (from which the hankel functions ultimately arise) is (Griffiths 4.37) $$ -\frac{\hbar^2}{2m} u''(r) + \left[V + \frac{\hbar^2}{2m}\frac{\ell(\ell+1)}{r^2}\right]u(r) = E u(r) $$ and itself involves $\ell$. In other words, once you choose a product solution of the form $R(r)Y_{\ell, m}(\theta, \phi)$, the radial equation necessarily involves that specific $\ell$ that appears in the product solution.