Introduction: The 1D-like Radial Problem
After separating angular dependence, the radial equation becomes effectively one-dimensional, with an effective potential that includes a centrifugal barrier.
The Radial Equation
\[\left[-\frac{\hbar^2}{2m}\frac{d^2}{dr^2} + \frac{\hbar^2 l(l+1)}{2mr^2} + V(r)\right]u(r) = Eu(r)\]where \(u(r) = rR(r)\) is the reduced radial wavefunction.
The Effective Potential
\[V_\text{eff}(r) = V(r) + \frac{\hbar^2 l(l+1)}{2mr^2}\]The centrifugal term \(\hbar^2 l(l+1)/2mr^2\) is repulsive, keeping particles with \(l > 0\) away from the origin.
Boundary Conditions
- \(u(0) = 0\) (so \(R(0)\) is finite)
- \(u(r) \to 0\) as \(r \to \infty\) (for bound states)
The Quantum Connection
The centrifugal barrier explains why s-orbitals (\(l = 0\)) can have non-zero probability at the nucleus while p, d, f orbitals (\(l > 0\)) vanish there. This affects chemistry: s-electrons experience the full nuclear charge and are more tightly bound.