Where the electron probably is

Ask where the electron is in a hydrogen atom and the honest answer is a curve rather than a number.

The most probable radius survives from the old model, and nothing else does. There is no orbit, no period, no speed at a given moment, and no trajectory to be interrupted. What replaces them is a distribution defined everywhere at once — a field of probability filling space, in the same sense the electric field fills space and for the same reason: a description in terms of something spread out turns out to be more useful than a description in terms of a thing at a place.

Why the peak sits at the Bohr radius #

The agreement is not a coincidence and it is not a survival of Bohr’s argument.

The 1s wavefunction falls off as e^(−r/a₀), so the probability density at a point is greatest at the nucleus and decreases monotonically outward. That sounds like it contradicts the figure, and it does not, because the figure is not plotting the density at a point.

The quantity drawn is the probability of finding the electron anywhere in a thin shell at radius r, which is the density multiplied by the shell’s volume, 4πr² dr. The density falls exponentially; the shell’s volume grows as r². Their product rises from zero, peaks, and falls — and the peak of 4πr²e^(−2r/a₀) is at exactly r = a₀.

So the Bohr radius appears as a competition between two opposite tendencies, neither of which is an orbit. Near the nucleus the electron is likely per unit volume and there is almost no volume; far out there is plenty of volume and almost no likelihood. One Bohr radius is where the two balance.

That factor of r² is the single most common source of confusion about atomic structure, and it is worth stating in its sharpest form: the electron is most likely to be found near the nucleus and most likely to be found at one Bohr radius, and both statements are true because they are about different questions.

What the shapes are counting #

Each curve is labelled by two whole numbers and both are doing something visible.

The principal number n fixes the energy — that is the ladder the spectrum reads — and roughly fixes the size, with the mean radius growing as n². The angular number l runs from 0 to n − 1 and fixes how much of the state’s motion is angular rather than radial.

The consequence is a rule about nodes: a state has exactly n − l − 1 radial nodes, radii at which the probability is zero. The 1s has none, the 2s has one, the 3s has two, the 2p has none, the 3d has none.

A node is not a place the electron passes through quickly. It is a place where the amplitude is exactly zero, so the electron is never detected there at all, and it is on the inside of the distribution rather than at its edge. The classical question — how does the electron get from one side of a node to the other — has no answer because it presupposes a trajectory.

Why there is a lowest state #

The ground state has an energy of −13.6 eV and there is nothing below it, which is the fact that makes atoms possible.

Classically there is nothing to stop an electron spiralling into the nucleus: the Coulomb potential goes to −∞ as r → 0, so there is no lowest energy, and an orbiting charge radiates, so the spiral should take about 10⁻¹¹ seconds. Every atom should have collapsed long ago.

The wave picture prevents it by the same argument that gives a confined particle a zero-point energy. Squeezing the electron into a smaller region shortens its wavelength, which raises its kinetic energy as 1/r², while the potential energy falls only as −1/r. The kinetic term wins at small radius, so the total energy has a minimum at a finite size, and that size is a₀.

The estimate is worth doing because it produces the right answer with no solving at all. Setting the kinetic energy to ħ²/2mr² and the potential to −e²/4πε₀r, then minimising the sum over r, gives r = 4πε₀ħ²/me² — the Bohr radius, 52.9 picometres, exactly. Atoms are the size they are because that is where confinement energy stops being worth paying.

How large an atom is, and why they are all the same size #

The distribution has a width, and the width is what “the size of an atom” means. It is worth taking seriously how odd the answer is.

Hydrogen’s mean radius is 1.5 a₀ = 79 picometres. Caesium, with fifty-five electrons and a nucleus fifty-five times as charged, has an atomic radius of about 265 picometres. Fifty-five times the charge and fifty-five times the electrons produce a factor of three in size, not a factor of fifty-five in either direction. Every atom in the periodic table sits between about 30 and 300 picometres, and the whole range is one order of magnitude.

The reason is the competition described above, running twice. Adding protons pulls the inner electrons in as 1/Z, so the innermost shell of a heavy atom is genuinely tiny — uranium’s 1s orbital has a mean radius of about a₀/92. But the outermost electron sees a nucleus screened by all the others, so the charge it experiences is a few units rather than Z, and it sits at a radius set by its own n² and that small effective charge. Size is decided by the outermost shell, and the outermost shell almost always sees a similar effective charge.

That near-cancellation is why chemistry is possible. If size grew with atomic number the way charge does, no two elements could form comparable bonds and there would be no periodic table to arrange them in.

What holds the shape up #

The balance can be seen from the other side too. Hydrogen is one charge sitting in the potential of another, and the potential is the same inverse-square field the site has already built.

The potential the electron lives in is a funnel steepening without limit toward the centre, and nothing about it changes between the classical and quantum treatments. What changes is what a particle can do in it: the classical electron slides down and the quantum one cannot, because the wavelength it would need at the bottom costs more kinetic energy than the descent releases. The ground state is where those two run out against each other, and the Bohr radius is the length at which they do.

That is why the funnel’s shape, rather than its depth, sets the character of the spectrum. A 1/r potential gives levels going as −1/n², crowding onto a limit; a square box gives n², spreading apart; a harmonic well gives evenly spaced ones. Reading a level pattern backwards to a potential is one of the standard moves in the subject, and it works because the map is so rigid.

Naming the approximation, twice over #

Two idealisations underlie every curve on this page and both are visible in the data.

The nucleus is not infinitely heavy. The formulas above treat the proton as fixed. It is not: both particles orbit their common centre of mass, and the correction is to replace the electron’s mass with the reduced mass, which for hydrogen is smaller by one part in 1836. That shifts every level by 0.054 per cent, which is enormous compared with the precision of a spectrometer — it is how deuterium was discovered, as a faint companion line offset from every hydrogen line by exactly the amount the heavier nucleus predicts.

The treatment is non-relativistic. The electron’s characteristic speed is αc ≈ c/137, so relativistic corrections enter at order α² ≈ 5 × 10⁻⁵. That is the fine structure: every level with l > 0 splits, and states that the non-relativistic treatment says share an energy stop sharing it. The figure’s clean degeneracy between 3s, 3p and 3d is an artefact of ignoring a correction of one part in twenty thousand.

Where the picture stops #

The one-electron treatment is exact for hydrogen and for no other atom, and the failure is not gradual.

Helium has two electrons that repel each other, and there is no closed-form solution. What is done instead is to treat each electron as moving in an averaged potential — the nucleus plus the smeared-out charge of the others — and then correct. That average is a real approximation with a real failure mode: it discards the correlation between the electrons, the fact that each avoids the other’s instantaneous position, and recovering that is most of what quantum chemistry is.

The consequence for the level scheme is immediate and structural. The accidental degeneracy of hydrogen — 2s and 2p at the same energy — is destroyed, because an s electron penetrates closer to the nucleus and is screened less by the inner electrons than a p electron is. The ordering that results, s below p below d, is what gives the periodic table its shape, and it does not exist in hydrogen at all.

The distribution is not a cloud of charge #

There is one reading of the figure that is nearly right and worth separating from the right one, because the difference decides several real calculations.

The curve is often described as showing how the electron’s charge is smeared through space, and for many purposes that description gives the correct answer: the electrostatic potential outside a hydrogen atom really is what a charge distribution of that shape would produce, and X-ray diffraction really does measure something that behaves like a continuous electron density.

But it is not a smear of charge, and the place the difference shows is in what happens when a measurement is made. A detector placed at 3 a₀ either registers a whole electron or registers nothing; it never registers the fraction of an electron the density at that radius would suggest. The distribution is a distribution of outcomes, and it looks like a static charge cloud only because the electrostatic effects of an electron in a stationary state depend on the average and not on the individual outcomes.

The distinction becomes operational the moment two electrons are involved. Treating them as two overlapping charge clouds gives the mean-field approximation and gets bond energies wrong by several electronvolts, because two clouds do not avoid each other and two electrons do. Recovering that avoidance — correlation — is the whole difficulty of quantum chemistry, and it exists because the clouds are a picture of averages rather than a picture of the thing.

Every curve on this page belongs to one rung of a ladder, and the ladder is what a spectrum measures. The shapes and the energies are two readings of one solution — which is why an experiment that finds the right energies is evidence about the shapes, and why the shapes had to be right long before anybody could see one. Forty years separated the spectrum from the first image, and the spectrum was the stronger evidence.

The same solution with the masses changed #

The essay has already noted that the proton’s finite mass shifts every level by one part in 1836, through the reduced mass. Pushing that observation harder gives three quite different atoms out of one solution, and each of them is an experiment.

Positronium is an electron bound to a positron. The two masses are equal, so the reduced mass is half the electron’s, and every energy is halved and every radius doubled: the ground state sits at 6.8 eV instead of 13.6, and the atom is twice the size. It also annihilates, in about a tenth of a nanosecond or a tenth of a microsecond depending on how the spins are arranged, which makes it the cleanest available laboratory for testing the theory of the bound state — no nucleus, no structure, nothing but two point particles and the field between them.

Muonic hydrogen replaces the electron with a muon, two hundred times heavier. The radius shrinks by that factor, so the muon orbits a couple of hundred femtometres out rather than fifty thousand — and since the probability of being inside the proton goes as the inverse cube of the radius, the muon spends some ten million times more of its time overlapping the nucleus than an electron does. That makes its energy levels enormously more sensitive to how big the proton is, which is why the atom was built.

Antihydrogen is a positron bound to an antiproton, and by every symmetry anybody believes in it should have exactly hydrogen’s spectrum. Making it, cooling it, holding it in a magnetic trap and doing spectroscopy on it took twenty years, and the answer so far is that its principal transition matches hydrogen’s to two parts in a million million. That is a test of a symmetry rather than of an atom, and the atom is the instrument.

What the sensitivity bought, and the argument it started #

Hydrogen’s 1s-to-2s transition is the most precisely measured quantity in atomic physics. Because the upper state is long-lived and the transition is driven by two photons from opposite directions — so the Doppler shifts cancel — its frequency is known to about four parts in a thousand million million, some fifteen digits.

A number known that well is not really a measurement of hydrogen. It is a measurement of everything that goes into predicting it: the Rydberg constant, the electron-to-proton mass ratio, the corrections from the quantised electromagnetic field, and — because an s electron has a small but non-zero probability of being inside the proton — the proton’s own radius.

That last term is where the trouble started. Extracting the proton radius from ordinary hydrogen and from electron scattering gave about 0.88 femtometres for decades. In 2010 the muonic hydrogen measurement, ten million times more sensitive to the same term, returned 0.84 — a discrepancy of seven standard deviations between two numbers that had no business disagreeing, and one that could not be waved away as a bad measurement because the muonic result was by far the sharper of the two.

For most of a decade it was called the proton radius puzzle and taken seriously as a possible sign that muons and electrons do not couple identically. What settled it was not new physics: a series of fresh measurements on ordinary hydrogen, and a re-analysis of the scattering data, moved the electronic value down onto the muonic one, and the accepted radius is now the smaller number. The lesson is the ordinary one about a discrepancy between a precise result and an old consensus, and the sensitivity that produced it came entirely from putting a heavier particle into the same solution.

What the picture cannot show #

The most important thing missing is the angle. Every curve here is a radial distribution, summed over all directions, and it says nothing about the shape a chemist means by “a p orbital”. The 2p state is not spherically symmetric — it has two lobes and a nodal plane — and that information has been integrated away to draw a curve on a page.

The second is the phase. A wavefunction is complex, and its sign or phase is invisible in a probability. That sign is the whole content of chemical bonding: two atomic orbitals overlapping in phase make a bond and out of phase make an antibond, and the probability plot cannot tell those apart because squaring destroys the distinction.

The third is time. A stationary state’s distribution does not change, which the drawing shows correctly and misleadingly — it looks like a static cloud, and the correct reading is that nothing about the electron’s position is changing because nothing about the electron’s position is defined. The drawing is a description of what repeated measurements on identically prepared atoms would give, not a picture of a thing.

Seeing the shape rather than inferring it #

For most of the twentieth century the distribution was inferred: it reproduced the spectrum, the ionisation energy, the polarisability and the scattering cross-section, and that agreement was the evidence. Three developments have since made it something closer to an observation.

X-ray diffraction of the density. A crystal’s diffraction pattern is the Fourier transform of its electron density, so inverting it returns the density directly. The resolution is not enough to see one atom’s radial nodes, and it is enough to see the density accumulated between two atoms in a bond — the charge transfer that “a covalent bond” names, measured rather than modelled.

What makes an image possible at all is an exponential. Transmission through a vacuum gap falls by about an order of magnitude for every ångström added, so a tip a nanometre from a surface carries a current dominated overwhelmingly by its single closest atom — and an instrument with a hand-ground tip resolves structure finer than the tip is. The sharpness comes from the exponent rather than from the machining.

Scanning tunnelling microscopy. A tip held a nanometre above a surface passes a current that depends exponentially on the gap, so the tunnelling probability maps the local density of states with sub-atomic resolution. The images of individual molecular orbitals produced this way have nodes in them, in the places the calculation puts them.

Photoionisation tomography. Ionising a hydrogen atom in a static field and imaging where the electrons land produces a pattern that is the projection of the bound state’s own nodal structure, magnified by the field. Published images of hydrogen’s n = 2 and higher states show the node counts directly.

None of these is a photograph of an electron in an orbit, because there is no such thing to photograph. What they measure is exactly what the curve on this page plots — a distribution of outcomes — and finding the nodes where the theory puts them is a strong test precisely because a node is a place where the answer is zero, which no fitting parameter can fake.

Where the ladder goes next #

The rungs from here: the angular part, and where the familiar orbital shapes come from; the effective potential and the centrifugal barrier that keeps high-l states away from the nucleus; screening and penetration, and why the ordering of subshells is what it is; the variational principle, which is how the helium ground state is actually computed; hyperfine structure and the 21-centimetre line, which is a splitting so small it takes ten million years to decay and is nevertheless the most-observed line in radio astronomy; and the exclusion principle, which decides how many electrons each of these shapes may hold.

The claim to carry forward is what survived and what did not. The Bohr radius is real, measurable and correct; the orbit it was derived from does not exist. A model can produce the right number by the wrong mechanism, and the way that is discovered is not by checking the number — it is by asking the model a second question, which in this case was the shape of the distribution, and getting an answer that could be compared with a measurement.