A list of states, not a ramp

The scaffold gets three things right: an electron bound to a nucleus, circular motion, an attractive inverse-square force. It gets two things wrong the moment you push it, and both are worth naming.

1. A satellite can take any radius. An electron cannot. Only certain orbits are allowed, which is the whole content of the model.
2. An orbiting charge should radiate continuously and spiral in. In the model it does not. An electron in an allowed state radiates nothing at all.

So the picture of an electron drifting through the space between $n = 3$ and $n = 2$, giving off light the whole way down, describes something the model explicitly rules out. A transition is a jump between two states, and one photon carries $E_3 - E_2$.

Treat the allowed states as a list rather than a ramp. The orbit picture is scaffolding; the energy-level diagram is the part that keeps working, and it is the part every later topic in this unit uses.

For hydrogen:

$$r_n = n^2 \times 0.053\ \text{nm}, \qquad E_n = -\frac{13.6}{n^2}\ \text{eV}.$$

The radius grows as $n^2$; the energy magnitude falls as $1/n^2$. Swapping the two shrinks the atom as $n$ rises, or makes high levels deeply bound, and every downstream comparison then inverts: which transitions are energetic, which lines are short-wavelength.

Anchor both at $n = 1$ and read upward:

1. $n = 2$: four times farther out, at $-3.40$ eV.
2. $n = 3$: nine times farther out, at $-1.51$ eV.

The physical sentence that ties them together: farther out means more weakly bound. So as $n$ climbs, the atom keeps getting larger while the levels crowd together toward $0$ eV. That crowding is directly visible in a hydrogen spectrum, where the lines of a series bunch up toward the series limit.

Quantization is not a rule Bohr simply asserted. It follows from requiring the electron's de Broglie wave to close on itself around the orbit:

$$n\lambda = 2\pi r, \qquad \lambda = \frac{h}{p}.$$

Only a whole number of wavelengths fits a closed loop. A fractional fit arrives back out of step with itself on the next trip round and cancels, so that radius is not available.

Two conditions are working together, and both are needed:

1. The electric force supplies the centripetal force, which relates $r$ to $v$.
2. The closure condition selects which of those radii survive.

Without the second there is no reason for the atom to have a lowest state at all, and the electron would spiral in. It is the same reason a string fixed at both ends sounds only certain notes: the boundary requires the wave to fit.

It is worth being explicit about the scope, because the model's successes are what make its failures surprising.

What it gets right: the energy levels of hydrogen, to remarkable accuracy, and therefore the wavelengths of every line in the hydrogen spectrum. That was the result that made the model impossible to ignore.

What it does not: multi-electron atoms, the relative brightness of spectral lines, and the actual shape of an electron's distribution around a nucleus. The modern picture replaces definite orbits with probability distributions, and the electron has no trajectory at all.

For this course, the working content is the energy-level ladder and the transition rule. Keep those and the planetary drawing becomes what it is: a memory aid for a list of numbers.