Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

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: $$\psi(r,\theta,\phi)=Ae^{ikz}+A\sum\limits_{l,m,n}^{\infty}C_{l,m,n}h_{n}(kr)Y_{l}^{m}(\theta,\phi).$$

share|improve this question
add comment

1 Answer

up vote 3 down vote accepted

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.

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.