The radial equation and centrifugal barriers

For a constant reduced mass $\mu$ and central potential $V(r)$, choose simultaneous eigenfunctions of $H$, $L^2$, and $L_z$. Write $\psi=R_\ell(r)Y_{\ell m}(\theta,\phi)$ with the spherical harmonic normalized over solid angle. Its angular eigenvalues are $\hbar^2\ell(\ell+1)$ and $m\hbar$, with integer $\ell\ge0$ and $-\ell\le m\le\ell$. The radial equation depends on $\ell$ but not on $m$ when no direction-selecting interaction is present.

The volume element is $r^2dr\,d\Omega$. Defining $u_\ell=rR_\ell$ removes the first derivative from the radial kinetic operator and turns normalization into an ordinary half-line integral. The shell probability between nearby radii is $|u_\ell|^2dr$. This does not make $|R_\ell|^2dr$ the shell probability, nor does it discard the angular distribution for measurements in a particular direction. Each expression must be paired with its own measure.

The reduced equation contains $\hbar^2\ell(\ell+1)/(2\mu r^2)$ in addition to the physical potential. It is nonnegative and rises toward the origin for nonzero angular momentum. This centrifugal barrier comes from angular kinetic energy, not an extra interaction added to the original Hamiltonian. It suppresses low-energy penetration into small radii for higher partial waves. It does not imply that every effective potential has a positive classical turning radius for every energy.

For a potential regular at the origin, the leading radial solutions behave as $u\propto r^{\ell+1}$ and $u\propto r^{-\ell}$. The regular three-dimensional Hamiltonian selects the first. In particular, the s-wave condition is $u(0)=0$ even though $R(0)$ may be finite and nonzero. Square-integrability alone can fail to exclude a singular s-wave candidate; absence of a contact interaction and the operator domain also matter. Singular inverse-square interactions require a separate endpoint analysis.

For each allowed radial level at fixed $\ell$, the $2\ell+1$ magnetic labels have equal energy in a spinless central problem. This rotational degeneracy does not mean there are $2\ell+1$ different radial solutions. Conversely, states with different angular momentum need not share an energy in a generic central potential. Hydrogen's additional equality across several angular momenta is a special Coulomb feature, rather than a consequence of spherical symmetry by itself.

Bound radial states must decay sufficiently at infinity to have finite norm. Positive-energy partial waves instead describe scattering and use continuum normalization. An effective barrier may also support a resonance that leaks outward rather than a square-integrable stationary eigenstate. A peak trapped behind a barrier in a finite-time picture therefore does not establish a true bound state. The spectral distinction uses the endpoint conditions and energy threshold, not simply the presence of a hump in an effective-potential sketch.