Each value of $\ell$ gives its own radial equation, and each radial equation has its own ladder of solutions, counted by the number of nodes of $u$. So a state of a central potential is labelled by three numbers: the radial number, $\ell$ and $m$. The energy depends on the first two. And the half-line brings a surprise: in three dimensions a shallow well can have no bound state at all, because the $s$-wave equation is the odd half of a one-dimensional well.

A tall building has floors and, on each floor, rooms. $\ell$ picks the floor’s wing, the radial number the floor, $m$ the room — and all the rooms of one floor and wing are at the same height.

The lowest levels of a particle in a sphere with an impenetrable wall, in units of the lowest one, in columns for $\ell=0,1,2$. Within a column the levels are the radial ladder; the $p$ level holds 3 states and the $d$ level 5, one for each $m$. The energy depends on $$ as well as on the radial number.

For each $\ell$ the radial equation has solutions with $n_{r}=0,1,2,\dots$ nodes of $u$ between the origin and infinity. The state has energy , the same for all values of .

• For fixed $\ell$ the energies grow with $n_{r}$, and for fixed $n_{r}$ they grow with , since the centrifugal term is positive.

For $\ell=0$, $u$ obeys the one-dimensional equation on $r>0$ with $u(0)=0$. Extended to negative as an odd function, it is an odd state of the one-dimensional well of width . The finite-well lesson showed that the first odd state needs .

The radial equation for $\ell=0$: no centrifugal term.

Extending $u$ as an odd function gives a solution of the one-dimensional well on the whole line, since $V$ is even.

From the finite-well lesson: the odd condition first has a root at $z_{0}=\pi/2$.

Square both sides.

A positive term added to the potential can only raise the energies.

In three dimensions, the $s$-wave radial equation is the odd half of a one-dimensional well. A spherical well therefore binds only when it is deep and wide enough.

A state of a central potential is labelled by its radial number $n_{r}$ (the nodes of $u$), by $\ell$ and by $m$. The energy depends on $n_{r}$ and only.

A one-dimensional well of any depth binds at least one state. A three-dimensional well does not: too shallow or too narrow, and nothing is bound.

For hydrogen, the number in front of the letter is $n=n_{r}+\ell+1$, where $n_{r}$ counts the nodes of .

Great work! You now know how to:

• label states of a central potential by $n_{r}$, $\ell$ and $m$
• prove that a shallow spherical well binds nothing
• read spectroscopic notation and count states in a level