With the angles taken care of, the Schrödinger equation for a central potential becomes an equation for one function of $r$. Written for $u=rR$ it looks exactly like the one-dimensional equation on a half-line, with two changes: the potential gains the centrifugal term $\hbar^{2}\ell(\ell+1)/2mr^{2}$, and $u$ must vanish at the origin. Everything learned about wells and barriers on a line now applies to atoms and nuclei.

A marble rolling in a bowl can be described by its distance from the centre alone, once its angular momentum is fixed: the spinning part of its motion acts like an extra push outwards. The radial equation is that description.

How the radial function $R=u/r$ starts at the origin for $\ell=0,1,2$: as a constant, as $r$ and as $r^2$, since . Only the state is nonzero at the centre; the higher , the more the centrifugal term keeps the particle away.

Put $\psi=\frac{u(r)}{r}Y_{\ell m}$ into the Schrödinger equation with a central potential. The angles drop out and obeys a one-dimensional equation on .

• Normalisation: $\int|\psi|^{2}d^{3}r=\int_{0}^{\infty}|u|^{2}dr=1$, because and is normalised.

Split the Laplacian into its radial part and $-\hat{L}^{2}/\hbar^{2}r^{2}$. The spherical harmonic turns $$ into and cancels throughout. The radial part, applied to , is simply , and multiplying by leaves the equation for .

The Laplacian in spherical coordinates.

The angular part acts on the spherical harmonic alone.

Expand both sides: each is $R''+\frac{2}{r}R'$.

Cancel $Y_{\ell m}$ from every term.

Multiply by $r$.

Writing the radial function as $u(r)/r$ turns the radial part of the Laplacian into a plain second derivative, so $u$ obeys a one-dimensional equation.

The radial equation has the potential plus a centrifugal term $\hbar^{2}\ell(\ell+1)/2mr^{2}$.

The radial equation contains $\ell$ through the centrifugal term, but never $m$. So energies depend on $\ell$ and on the radial solution, not on $m$.

Near the origin the centrifugal term dominates, and $u$ grows as $r$ to the power $\ell+1$.

Great work! You now know how to:

• derive the radial equation for $u=rR$
• use the conditions at the origin and the normalisation of $u$
• read s states as one-dimensional problems on a half-line