No two in the same state, and why matter has volume

The energy levels of an atom are computed for one electron. Nothing in the calculation says what happens when a second is added, and the obvious guess — that both settle into the lowest level, since that is where the energy is least — is wrong for every element after hydrogen.

If every electron went to the bottom, every atom would be a small dense ball with one occupied level, all elements would behave alike, there would be no chemistry, and nothing would resist being squeezed. The rule that prevents it is one sentence long and is not derived from any force.

The rule #

No two identical fermions may occupy the same quantum state. For electrons in an atom, a “state” is a full specification — which orbital, and which of the two spin outcomes — so each orbital holds exactly two electrons, one of each spin.

Pauli proposed it in 1925, as an Ausschließungsregel, to account for the pattern of spectral lines and for the lengths of the periods in the chemical table. He had no mechanism, disliked the lack of one, and described it as something he could not derive.

The derivation came later and it is a statement about identity rather than about forces. Two electrons are not merely similar; they are indistinguishable in a strong sense — there is no property that could label one of them, and swapping them produces a state that must give the same predictions. That leaves the total wavefunction free to change by at most a sign under the swap, and the two possibilities define the two families of particle in nature.

Particles whose wavefunction changes sign are fermions. Ones whose wavefunction is unchanged are bosons. And a sign-changing wavefunction for two particles in the same state is equal to minus itself, so it is zero: the configuration does not exist. The exclusion principle is that piece of arithmetic.

What the rule is not #

It is worth being precise about three things it is not, because all three are commonly asserted.

It is not a force. No potential energy term is added anywhere. Two electrons repel each other electrostatically, which is a separate and much weaker effect at atomic scales; the exclusion is a constraint on which states may be occupied, and constraints have consequences without exerting forces in the mechanical sense.

It is not a rule about position. Two electrons may be in the same place. What they may not do is occupy the same state, and states are specified by their whole description — an electron in a 1s orbital with spin up and another in the same orbital with spin down overlap completely in space and are in different states.

It does not apply to everything. Photons are bosons and pile into one state without limit, which is what a laser is. Helium-4 atoms are bosons and condense; helium-3 atoms are fermions and do not, until they pair up. Two isotopes of the same element, differing by one neutron, behave in categorically different ways at low temperature because of a sign.

The shells, and the shape of the table #

Applying the rule to the hydrogen levels produces the periodic table’s structure, and the arithmetic is short.

The level n — the one the ladder of hydrogen counts — contains orbitals with l running from 0 to n − 1, and each l has 2l + 1 orientations. Summing gives n² orbitals, and doubling for spin gives 2n² states: two in the first shell, eight in the second, eighteen in the third.

That gives periods of 2, 8, 18, 32 — and the observed periods are 2, 8, 8, 18, 18, 32. The discrepancy is real and its cause is worth naming: in a multi-electron atom the 4s level lies below the 3d, because an s electron penetrates inside the screening cloud and feels more of the nuclear charge. So the fourth period fills 4s, then 3d, then 4p, and comes out eight-plus-ten long rather than thirty-two.

The exclusion principle supplies the capacities; the ordering comes from screening, which is a consequence of the electrons’ mutual repulsion. Both are needed and neither alone gives the table.

Why a table holds a cup up #

The mechanical consequence is the one that is hardest to believe and easiest to check.

Push two atoms together. Their electron clouds overlap, and the electrons in the overlap region would have to occupy states that are already filled. The only available states are higher ones, so the energy rises steeply — and a steeply rising energy with separation is a repulsive force.

That force is what stops a hand passing through a table, and it is not electrostatic repulsion. An atom is neutral, and at the separations involved the electrostatic term is comparatively weak; what dominates is the cost of finding unoccupied states. The everyday hardness of matter is a counting rule.

The same argument scaled up gives degeneracy pressure. Compress a gas of electrons and each is confined more tightly, so each state’s energy rises as 1/L², and because the low states are all occupied the average energy rises with them. The resulting pressure exists at absolute zero and does not vanish with temperature — which is a genuinely strange kind of pressure, since it comes from the exclusion principle plus confinement and not from thermal motion at all.

Why a table holds a cup up is the same counting seen as a force. Four occupied states in one box each contribute their own zero-point energy, and the highest is the most expensive. Squeeze the box and all four rise together — the lowest by the least, the highest by the most — and the total is what pushes back. Nothing in that description is a force between particles. It is four energies that all increase when the walls come in, and the resistance to being compressed is the derivative of their sum.

The same rule, at astronomical scale #

The place degeneracy pressure is not a subtlety is in the interiors of dead stars, and the fleet’s astronomy site owns that argument in detail. Two features of it belong here because they are the exclusion principle rather than the astrophysics.

The first is that the pressure depends on density and not on temperature. A star supported by ordinary gas pressure cools and contracts; one supported by degeneracy pressure cools without contracting at all, because the support was never thermal.

The second is that the support has a limit, and the limit comes from relativity rather than from the counting. As the density rises the electrons’ momenta approach mc, the relation between their energy and momentum changes from p²/2m to pc, and the pressure stops rising fast enough to keep up with gravity. That is where a mass ceiling comes from, and it is one of the few places in physics where a rule about identity and a rule about speed limits combine to produce a number.

What happens when the rule is relaxed #

The complementary case makes the principle visible by its absence.

Bosons have no exclusion, and at low enough temperature a macroscopic fraction of them occupies the single lowest state. Liquid helium-4 below 2.17 K flows without viscosity and climbs the walls of its container; a dilute gas of rubidium atoms below a microkelvin forms a condensate whose entire population shares one wavefunction. Neither behaviour has any counterpart for electrons.

The comparison that makes the point is helium-3 against helium-4. Chemically identical, differing by one neutron, and therefore differing in whether the atom as a whole is a fermion or a boson. Helium-4 becomes superfluid at 2.17 K. Helium-3 does not, until 2.5 millikelvin — a thousand times colder — and then only because its atoms pair up, and a pair of fermions is a boson. One neutron, three orders of magnitude in temperature.

Counting the states, and the numbers that come out #

The arithmetic is worth doing once in full, because the numbers it produces are recognisable and were known as chemical facts long before there was a reason for them.

Hydrogen has one electron: 1s¹, and one unpaired electron makes it reactive. Helium has two: 1s², a filled shell, and helium reacts with nothing. Lithium has three, so the third must start a new shell — 1s²2s¹ — and lithium behaves like hydrogen because it also has one loosely held electron outside a closed core. Neon closes the second shell at ten, argon the third at eighteen.

That the pattern repeats — that element 3 resembles element 11 resembles element 19 — was Mendeleev’s observation in 1869 and had no explanation for fifty-six years. What supplies it is the capacity 2n², which is a counting result, together with the fact that only the outermost shell is chemically accessible.

The lengths of the periods are the same counting read as a sequence: 2, then 8, then 8 again because 3d waits for 4s, then 18, then 18, then 32. Every irregularity in that list is a screening effect, and every regularity is the exclusion principle.

What a “state” has to include #

The count of two per orbital was not obvious in 1925, and the history of getting it right is a good illustration of what the word “state” has to cover.

Bohr and Sommerfeld’s model gave each level a set of orbits labelled by two quantum numbers, and counting those gave n² — half the observed capacity. Pauli’s response was to postulate a fourth quantum number with two values, described as “a peculiar, classically indescribable two-valuedness”, and to state the exclusion rule in terms of the four together. He did not say what the fourth number was.

Uhlenbeck and Goudsmit identified it as spin later the same year, and the capacity became 2n² with a physical reason behind the factor of two.

The lesson is a general one about exclusion arguments: the rule is only as good as the enumeration of states it is applied to. Getting a capacity wrong by a factor of two is evidence that a degree of freedom has been missed, and that is how spin was found — not by observing something spinning, but by counting places and finding twice as many occupants as places.

Where the shell picture stops #

The tidy account above is a model with a stated domain, and three of its assumptions fail in ways that matter.

Electrons are not assigned to orbitals. The exact state of a multi-electron atom is not a product of one-electron orbitals; it is an antisymmetrised combination, and configurations mix. “Carbon’s outer electrons are in 2s²2p²” is a leading term, not a description, and for transition metals the leading term is sometimes wrong about the ground state.

The ordering is not fixed. The 4s-below-3d rule is true for potassium and calcium and stops being true once the 3d shell starts filling — in every transition metal ion, 3d is below 4s. Textbook orderings drawn as a fixed staircase are describing one atom’s arithmetic and presenting it as a law.

Relativity reorders the heavy elements. In a heavy atom the innermost electrons move at an appreciable fraction of c, which contracts the s orbitals and expands the d and f ones. Gold’s colour and mercury’s liquidity at room temperature are both consequences, and neither is derivable from the non-relativistic shell picture.

The rule at 10²³, which is a different regime #

Applying the principle to two electrons gives chemistry. Applying it to a mole of them gives a quantity with no atomic counterpart.

In a metal the conduction electrons fill states from the bottom up, and the energy of the highest occupied one is the Fermi energy — several electronvolts, which corresponds to a temperature of tens of thousands of kelvin. So the electrons in a room-temperature copper wire are not a thermal gas at 300 K; they are a nearly full sea whose typical member has an energy a hundred times larger than kT, put there by the exclusion principle and not by heat.

That resolves a puzzle that stood for thirty years. Classical theory predicted that each conduction electron should contribute (3/2)k to a metal’s heat capacity, which would make metals’ heat capacities far larger than they are. The measured electronic contribution is smaller by about a factor of a hundred, and the reason is that heating a metal by a few kT can only promote the electrons within kT of the top of the sea — a tiny fraction of them. The rest have nowhere to go.

Where the shell picture stops is where the atoms stop being separate. Bring many together and each sharp level splits into as many nearly-degenerate ones as there are atoms, because the exclusion principle forbids them to coincide — and a mole of atoms turns a level into a band so dense that it is continuous for every practical purpose. The rule has not changed; only the number of things it is being applied to.

The general shape is worth keeping. At two particles the principle is a selection rule; at 10²³ it is a pressure and an energy scale, and both are properties that no amount of studying one electron would suggest.

Why matter is extensive, which is a theorem #

The title of this essay is a claim, and it turns out to be a theorem — one that took forty years after Pauli to prove, and whose proof needs the exclusion principle and cannot be done without it.

The question is this. Take N electrons and enough nuclei to balance them, with nothing but the Coulomb interaction between every pair. Is the lowest energy the system can have bounded below by a constant times N? If it is, matter is extensive: twice as much of it has twice the energy and twice the volume, and bringing two lumps together releases nothing dramatic. If it is not, none of that holds.

It is not obvious that the answer is yes. The Coulomb energy between an electron and a nucleus goes to minus infinity as they approach, so there is a bottomless well at every nucleus, and with N particles and N nuclei there are N² attracting pairs against N² repelling ones. The uncertainty principle alone rescues a single hydrogen atom, by the argument three sections above — but it does not obviously rescue a mole of them, because nothing in it stops every electron from crowding into the same favourable region.

Dyson and Lenard proved in 1967 that the answer is yes for electrons, and Lieb and Thirring gave a much sharper proof eight years later. What makes the proof work is that the exclusion principle forces the electrons into successively higher kinetic-energy states as they are packed together, so the kinetic cost of crowding rises faster than the electrostatic gain.

Dyson also worked out what happens if the principle is removed, and the answer is worth stating precisely because it is not “slightly different”. With bosonic electrons the ground-state energy would go as $-N^{7/5}$ rather than as $-N$. That is not extensive: bringing two lumps of such matter together would release an energy growing faster than their mass, so any two objects would merge explosively and no stable bulk substance could exist at all.

So the everyday facts that a kilogram of anything occupies a predictable volume, and that pushing two bricks together does nothing, are consequences of a sign in a wavefunction — established as a theorem rather than assumed.

Looking for the violation #

A rule this consequential invites a search for exceptions, and the search has a rather beautiful design.

Suppose the principle were very slightly violated — that an electron could, with tiny probability, enter a shell that is already full. In a copper atom the 1s shell holds two electrons; a third, arriving where two already sit, would be more screened than an ordinary one and would emit an X-ray of slightly lower energy on settling. The normal copper line is at about 8.04 kilo-electronvolts; the forbidden one would be some three hundred electronvolts below it, in a region of the spectrum where nothing else appears.

So the experiment is: pass a large current through a copper strip, so that electrons which have never been part of those atoms are continuously introduced, and look for a peak where no peak should be. The current matters because the exchange symmetry of a given set of particles cannot change — only new electrons, which have not previously been antisymmetrised with the others, can test it.

Nothing has been found. Successive versions of the experiment, most recently deep underground where the cosmic-ray background is suppressed, have pushed the bound on the probability of such a transition below one part in $10^{29}$, and further improvements are in progress.

That is a stricter limit than almost any other statement in physics enjoys, and it is worth noticing what kind of statement is being tested. Not a numerical constant, and not a force law: a symmetry — whether the wavefunction of two identical particles really does change sign, exactly, every time.

What the picture cannot show #

The opening figure draws occupied levels as heavier lines, which is a bookkeeping device and not a picture of anything. There is no sense in which one electron is “on” a level; the state of N electrons is a single antisymmetric function of all their coordinates, and drawing it as N separate occupancies is the approximation being made rather than the physics being described.

The figure also cannot show the sign. Everything on this page follows from a minus sign under exchange, and a minus sign has no visual representation in a diagram of energies. That is why the rule looks like a stipulation when it is drawn and like a theorem when it is written.

And nothing here shows that the rule is exact. It is not a strong tendency or a good approximation; no violation has ever been observed, and experiments searching for one have bounded the probability of a symmetric two-electron state at below 10⁻²⁹. Few statements in physics are tested that hard.

Where the ladder goes next #

The rungs from here: the spin–statistics theorem, which derives the connection between spin and exchange symmetry from relativity and locality rather than assuming it; Hund’s rules, and why a half-filled shell is unusually stable; the Fermi sea and the Fermi energy, which is the same counting done for 10²³ particles; superconductivity, where fermions pair into bosons and the exclusion is evaded rather than broken; and band structure, where filling decides whether a material conducts.

The claim to carry forward is where the rule comes from. It is not a force and not an observation about crowding — it is what indistinguishability costs. Two objects that cannot be told apart even in principle constrain the mathematics that describes them, and the constraint turns out to be why atoms have structure, why the periodic table has rows, and why solid things are solid.