Physics · Module 22
Module 7 showed you that a string fixed at both ends can only sing certain notes. Module 21 showed you that an electron travels as a wave. Put those two facts together and you get the structure of every atom and the colour of every flame.
Before any quantum mechanics at all, remember what you already know from Module 7.
Take a string, fix both ends, and pluck it. A travelling wave runs off along the string, hits the fixed end, reflects, comes back, meets the wave still going out, and the two add together. For almost every wavelength this is a mess — the returning wave arrives out of step with the outgoing one and they mostly cancel, leaving nothing. But at certain special wavelengths the returning wave arrives exactly in step, reinforces, and a large, stable pattern builds up: a standing wave.
Which wavelengths? The ones that fit. The string is held still at both ends, so the pattern must have a node there, and that allows exactly
Note carefully where the whole numbers came from. Nobody put them in. There is no rule anywhere saying "waves must come in whole numbers". The integers appeared because we confined a wave — we trapped it between two walls — and only some wavelengths survive being trapped. Confinement produces integers. That sentence is the entire bridge from Module 7 to the atom.
Now, Module 21 told you that an electron travels as a wave of wavelength λ, where λ = h/p. So take an electron and trap it. The bench below writes that wave as ψ (Greek psi); its second view shows |ψ|², the size of ψ squared, which is what tells you the probability of finding the electron at that point — Module 21's amplitude-squared rule, applied here.
Look at the probability view, |ψ|², and notice something that has no classical counterpart at all. In the n = 2 state there is a point in the middle of the box where the electron is never found — and yet it is found on both sides of that point. A classical particle bouncing between two walls would have to pass through the middle to get from one side to the other. This one does not, because it is not a little ball travelling about; it is a standing wave with a node there, and the node is a place of zero probability.
You take the same box and make it four times as wide.
What happens to the energy of the lowest rung, and to the gaps between rungs?
Sixteen times smaller, because E ∝ 1/L². This single relationship is doing a great deal of work in the world. It is why quantum dots — nanocrystals a few nanometres across — glow a colour that depends only on their size: the same material, made into smaller dots, emits bluer light, because a smaller box means a wider energy ladder means a bigger jump means a higher-frequency photon. QLED televisions are sold on exactly this equation.
And it is why confinement is expensive. Squeezing a quantum object into a smaller space forces its energy up, hard. Remember that; in section 23.2 it turns out to be the reason atoms have a size at all.
An atom is not a box with square walls. It is an electron trapped by the electrical attraction of a nucleus — the Coulomb attraction of Module 13, pulling inward, getting weaker with distance. But it is still a trap, and a trapped wave still has to fit.
Niels Bohr, in 1913, guessed at the answer before de Broglie had supplied the reason. Take the electron to be going round a circle of radius r. For the wave to survive going round and round, it must join up with itself smoothly after one lap — otherwise it interferes with itself destructively and dies. So a whole number of wavelengths must fit round the circumference:
Now combine that with Coulomb's law as the centripetal force of Module 6 — two equations in two unknowns — and the allowed radii and the allowed energies all fall out in half a page of algebra:
Now the payoff. The electron can only have those energies. When it drops from one rung to a lower one, the energy it gives up is exactly the gap between the two rungs — no more, no less. By Module 21, it does that by emitting one photon with precisely that energy, at a frequency set by E = hf. So a hydrogen atom cannot emit any colour it likes; it can only emit a particular set of colours, and the set is calculable.
Drop from n = 3 to n = 2 and the photon carries 13.6(1/4 − 1/9) = 1.89 eV; and since E = hf with c = fλ, that gives λ = hc/E ≈ 656 nm — a deep red. Hold a hydrogen discharge tube up to a prism and there it is: a sharp red line, at 656.3 nm, exactly where the arithmetic says.
Play with the top slider first. At 3.0 wavelengths the wave meets itself perfectly after one lap. At 3.4 it does not — the ends do not match, and where they overlap you have a crest landing on a trough. That state simply cancels itself out and does not exist. The whole numbers in the atom are exactly the whole numbers on the guitar string, and for exactly the same reason.
Then work the transition sliders and read off the lines. Every jump ending at n = 2 lands in or near the visible band — that is the Balmer series, and it is what you see in a hydrogen tube: red at 656, blue-green at 486, violet at 434 and 410. Every jump ending at n = 1 is far in the ultraviolet (the Lyman series). Everything ending at n = 3 or above is infrared.
Sunlight, spread out by a prism into a spectrum, is crossed by hundreds of narrow dark lines at very precise wavelengths. Two of the strongest sit in the yellow, at 589.0 and 589.6 nm — exactly the pair of wavelengths a sodium street lamp emits.
What is happening?
An atom absorbs at exactly the wavelengths it emits, because the same ladder of levels governs both directions: a photon can be swallowed only if its energy matches a gap. So cool gas lying in front of a hot source removes precisely those colours, and you get dark lines on a bright background — the same lines, in the same places, as the bright lines that gas would emit if you heated it.
This is arguably the single most useful fact in all of science. Every element has its own pattern of lines, as individual as a fingerprint, fixed by its own energy ladder. So you can point a spectrograph at a star, or a nebula, or a planet's atmosphere, or a distant galaxy, and read off what it is made of — without going there, and without any sample ever touching an instrument.
Helium was discovered this way. In 1868 astronomers found a yellow line in the Sun's spectrum that matched no known element; they named it after helios, the Sun. It was not found on Earth for another twenty-seven years. And when the same familiar patterns turn up in a distant galaxy shifted bodily towards the red, the size of that shift tells you how fast the galaxy is receding — which is how we know the universe is expanding. All of it rests on the fact that a trapped wave has to fit.
A sodium street lamp is orange because sodium's easiest transition emits at 589 nm and almost nothing else — which is why everything under one looks grey: there is no other colour in the light for objects to reflect. Neon signs are red-orange because that is neon's pattern; the other colours in "neon" signage are different gases, or coatings. The yellow flare when salt water boils over onto a gas ring is the same 589 nm sodium line. And a firework's colour is chemistry chosen for its energy ladder: strontium for red, barium for green, copper for blue.
Meanwhile a candle flame is a smooth continuous glow with no lines at all, because its light does not come from isolated atoms jumping between levels. It comes from hot soot particles — solid lumps with so many overlapping levels that the ladder blurs into a continuum. That is the blackbody radiation of Module 8, and Module 21's box on Planck.
Bohr's picture gets hydrogen's energies exactly right and is, as a picture, wrong. There are no orbits. The electron is not a small object going round a track, and drawing atoms that way — as almost every logo and textbook cover does — is a hangover from 1913.
The reason is Module 21's double slit. An electron does not have a position it merely keeps secret. Between preparations and measurements it does not have a trajectory at all. What it has is a wave, spread over space, whose squared size at each point gives the probability of finding it there if you look.
In 1926 Erwin Schrödinger wrote down the equation that such a wave has to satisfy, and solved it for the hydrogen atom. The solutions are called orbitals. They are three-dimensional standing waves in the nucleus's electrical trap — the direct descendants of the string in Module 7 — and instead of a circle you get a cloud, thickest where the electron is most likely to be found.
Three whole numbers come out instead of one, because there are three dimensions to fit a wave into:
| Number | What it counts | What it controls |
|---|---|---|
| n = 1, 2, 3, … | Counts how many half-waves fit into the wave's outward, radial direction — plus the angular ones counted by l. In Bohr's picture n counted whole wavelengths going round a circle; here it counts everything, radial and angular together, and the energy it gives is the same energy Bohr got. | Energy and overall size. |
| l = 0 … n−1 | How many angular nodes the cloud has | Shape. l = 0 is s (a sphere), 1 is p (two lobes), 2 is d (four lobes for four of the five d orbitals; the one drawn here, dz², is two lobes with a doughnut round the middle) |
| m = −l … +l | Which way the shape points | Orientation in space |
Bohr's n and Schrödinger's n are the same number and give the same energies, but they count different things, because Bohr was counting round a loop that is not there.
The next bench does not draw the cloud; it builds one, dot by dot. It picks a point in space at random, works out |ψ|² there, and keeps the point that often — a point where |ψ|² is twice as large is kept twice as often. Do that a few thousand times and the dots pile up exactly where the electron is likely to be. So what you are looking at is a real sample of where the electron would be found, not an artist's impression of it. Then it measures the average distance of those points from the nucleus, and you can check it against the number the theory predicts.
Two things to notice. First, the 1s cloud is densest right at the nucleus and thins outward — the electron's single most likely position is at the centre. Yet the histogram of "probability of being at distance r" peaks at 0.053 nm, the Bohr radius, because a thin spherical shell at radius r sweeps a volume proportional to r², and that weight of room more than compensates for the lower density. Both statements are true; they are answers to different questions.
Second, look at the nodes — the surfaces where the probability is exactly zero. In 2s there is a spherical shell where the electron is never found, with cloud on both sides of it. Same puzzle as the box, same answer: the electron is a standing wave, and standing waves have nodes.
Add more electrons and one further rule applies, due to Pauli: no two electrons in an atom may occupy the same state. So they fill the orbitals from the bottom up, two per orbital (the two allowed spin directions).
Count the orbitals available at each level and you get the shape of the periodic table straight off. n = 1 has only an s orbital: 2 electrons, so the first row has two elements. n = 2 has one s and three p: 8 electrons, so the second row has eight. The ten transition metals in each long row are the five d orbitals. The fourteen lanthanides are the seven f orbitals. The whole layout of the chart on the chemistry lab wall — its strange width, its blocks, the island of rare earths at the bottom — is a picture of which standing waves fit around a nucleus. Chemistry is a branch of this module.
| Idea | What it says |
|---|---|
| Confinement makes integers | A trapped wave can only take the wavelengths that fit — Module 7's string, unchanged. That is where all the whole numbers in quantum mechanics come from. |
| Particle in a box | En = n²h²/8mL². Discrete rungs, a non-zero lowest rung, and a ladder that stretches as 1/L². Quantum dots are sold on that. |
| The atom | A trapped wave must fit the trap. Bohr got the right answer by making a whole number of wavelengths close round a circle; there is no circle, but the fitting condition survives and gives En = −13.6/n² eV for hydrogen. |
| Spectra | A jump between rungs emits one photon of exactly that energy gap. Each element has its own line pattern, and absorption removes the same lines it would emit. This is how we know what stars are made of. |
| Orbitals | Not orbits: three-dimensional standing waves, with three whole numbers n, l, m. Fill them two at a time and you have the periodic table. |