Appearance
7.7 — Spin, the Exclusion Principle and Why Matter Takes Up Space
Chapter 7.6 produced three quantum numbers and they are not enough. Two facts prove it.
Fact one. Send a beam of silver atoms through a magnetic field that varies across the beam. The beam splits into two. Angular momentum \ell gives 2\ell+1 states — 1, 3, 5, 7 — always odd. Two is not on the list.
Fact two. With only three quantum numbers, every electron in every atom would sit in the 1s ground state, since that is the lowest energy and nothing forbids it. Every element would be chemically identical, there would be no periodic table, no chemistry, and no structure of any kind.
Something is missing, and it is the property that makes matter solid.
Stern–Gerlach
Otto Stern and Walther Gerlach performed this in Frankfurt in 1922. An atom with a magnetic moment \vec{\mu} in a non-uniform field feels a force:
F_z = \mu_z\frac{\partial B_z}{\partial z}
A uniform field would only rotate it (Chapter 4.5). The gradient is what pushes it, and by how much depends on the component of the moment along the field.
Classical prediction: the moments point in all directions, so \mu_z takes all values between -\mu and +\mu, and the beam smears into a continuous band.
Prediction from \ell alone: an odd number of spots, and for silver, whose outer electron is in a 5s state with \ell = 0, exactly one spot.
Result: two spots.
There is a good story here. Stern and Gerlach could not see the deposit on the detector plate until Stern breathed on it — he smoked cheap cigars, and the sulphur in the smoke converted the silver to black silver sulphide, developing the image. Both were poorly paid enough that the cigars were cheap enough to contain the sulphur.
They did not know what they had found. Electron spin was not proposed until 1925.
Spin
In 1925 two graduate students in Leiden, George Uhlenbeck and Samuel Goudsmit, proposed that the electron has an intrinsic angular momentum of its own — as though it were spinning.
s = \frac{1}{2}, \qquad m_s = \pm\frac{1}{2}
Two values, which is exactly the two spots. The number of states is 2s+1 = 2.
The magnitude:
|\vec{S}| = \sqrt{s(s+1)}\,\hbar = \frac{\sqrt{3}}{2}\hbar = 0.866\hbar
The z-component:
S_z = \pm\frac{\hbar}{2}
usually called spin-up and spin-down.
It is not spinning
The name is a historical accident and the picture is wrong. Three reasons, and each is decisive.
A spinning ball cannot do it. Take the classical electron radius, 2.8\times10^{-15} m, and ask how fast the surface must move to give angular momentum \hbar/2. The answer exceeds the speed of light by a factor of about 200. Make the electron smaller — and experiment says it is smaller than 10^{-18} m, possibly a true point — and the required speed grows.
Half-integer angular momentum is impossible for orbital motion. Chapter 7.6 derived that m_\ell must be an integer, because a wave has to join up with itself after going round once. Spin does not obey that constraint, so it is not motion in space.
Rotating a spin-½ particle by 360° does not return it to its original state — it multiplies the wavefunction by -1. You need 720° to get back. Nothing that rotates in ordinary space behaves this way.
Uhlenbeck and Goudsmit tried to withdraw their paper after Lorentz pointed out the speed-of-light problem. Their supervisor Ehrenfest had already sent it, and told them they were young enough to afford a stupidity.
What spin actually is: an intrinsic property of a particle, like charge and mass, that behaves mathematically like angular momentum and can be added to orbital angular momentum. It genuinely contributes to the total angular momentum of an atom, and it genuinely produces a magnetic moment. It just is not rotation.
Dirac's 1928 relativistic equation for the electron produced spin-½ automatically, without being put in — the first indication that spin is a consequence of combining quantum mechanics with relativity rather than an extra ingredient.
The magnetic moment and the anomaly
Spin gives the electron a magnetic moment:
\vec{\mu}_s = -g_s\frac{e}{2m_e}\vec{S}
where g_s is the g-factor. For orbital angular momentum, g_\ell = 1 exactly. For spin, classical reasoning also predicts 1.
Measured: g_s = 2.00231930436.
The 2 is explained by the Dirac equation, which gives exactly 2 with no adjustment. That was a triumph.
The 0.00231930436 is the anomalous magnetic moment, and it comes from quantum electrodynamics — the electron interacting with the fluctuating vacuum (Chapter 7.9). Schwinger computed the leading correction in 1948 and got \alpha/2\pi = 0.001161, which is most of it. Higher-order terms, involving thousands of Feynman diagrams computed over decades, bring theory and experiment into agreement to twelve significant figures.
g/2\ \text{theory} = 1.00115965218161
g/2\ \text{experiment} = 1.00115965218059
This is the most precisely verified prediction in the history of science. Feynman's comparison: it is like measuring the distance from New York to Los Angeles to within the thickness of a human hair.
And it is currently the sharpest place to look for new physics. The muon's anomalous moment, measured at Fermilab, sits a few standard deviations from the Standard Model prediction, and whether that gap is real is one of the liveliest questions in particle physics (Chapter 8.7).
The fourth quantum number
The complete set for an atomic electron:
| Symbol | Name | Values |
|---|---|---|
| n | Principal | 1, 2, 3, … |
| \ell | Orbital | 0 to n-1 |
| m_\ell | Magnetic | -\ell to +\ell |
| m_s | Spin | \pm\frac{1}{2} |
The Pauli exclusion principle
Wolfgang Pauli, in 1925, proposed the rule that makes matter possible:
No two electrons in an atom can have the same set of all four quantum numbers.
Immediate consequence: each orbital holds at most two electrons, one with each spin.
And immediately, the periodic table. Counting states per shell:
2\sum_{\ell=0}^{n-1}(2\ell+1) = 2n^2
giving 2, 8, 18, 32 — the row lengths of the periodic table, derived. Chapter 9.3 works through the filling order and the exceptions.
Pauli did not like it either. He called it "nonsense" that he had been forced into, and could give no reason for it until the deeper explanation arrived.
Identical particles and the deeper reason
The exclusion principle is a consequence of something more fundamental about what "identical" means in quantum mechanics.
Two electrons are not merely similar. They are indistinguishable in principle. There is no property whatsoever that differs, no serial number, no way even conceptually to label one and follow it. Classical objects can always be tracked by their trajectory; quantum particles have no trajectory (Chapter 7.2).
Consequence. Take a two-particle wavefunction \psi(1,2) and swap the labels. Since nothing physical changed, the probability density cannot change:
|\psi(2,1)|^2 = |\psi(1,2)|^2
So the wavefunction can change by at most a phase:
\psi(2,1) = e^{i\alpha}\psi(1,2)
Swap twice and you must be back where you started, so e^{2i\alpha} = 1, and:
e^{i\alpha} = \pm1
Exactly two possibilities, and nature uses both.
\boxed{\psi(2,1) = +\psi(1,2)\ \text{(bosons)}, \qquad \psi(2,1) = -\psi(1,2)\ \text{(fermions)}}
Where exclusion comes from
Fermions, with the minus sign. Suppose two of them are in the same state, so \psi(1,2) = \psi(2,1) trivially. Combined with antisymmetry:
\psi = -\psi \quad\Longrightarrow\quad \psi = 0
The wavefunction vanishes identically. The state does not exist.
That is the exclusion principle, and it is not an extra rule. It is arithmetic, following from antisymmetry.
Bosons, with the plus sign, have no such restriction. In fact they prefer to share: the symmetric combination gives them an enhanced probability of being in the same state, which is why lasers work (Chapter 5.6) and why Bose–Einstein condensates form.
The spin–statistics theorem
Which particles are which?
\boxed{\text{Half-integer spin} \Rightarrow \text{fermion}. \qquad \text{Integer spin} \Rightarrow \text{boson}.}
| Particle | Spin | Type |
|---|---|---|
| Electron, proton, neutron, quark, neutrino | ½ | Fermion |
| Photon, gluon, W, Z | 1 | Boson |
| Higgs | 0 | Boson |
| Graviton (hypothetical) | 2 | Boson |
| Helium-4 nucleus | 0 | Boson |
| Helium-3 nucleus | ½ | Fermion |
Pauli proved this in 1940 from relativistic quantum field theory, and the proof is genuinely difficult — the assumptions are that the theory be relativistic, that energies be bounded below, and that measurements at spacelike separation commute. Feynman, who could explain almost anything simply, wrote that he had never found a way to explain the spin–statistics theorem at an elementary level, and that this meant it was not fully understood.
The composite rule: count the constituent fermions. Even number gives a boson, odd gives a fermion. Helium-4 has 2 protons, 2 neutrons and 2 electrons — six fermions, so it is a boson. Helium-3 has one fewer neutron — five fermions, so it is a fermion.
This tiny difference has enormous consequences. Helium-4 becomes a superfluid at 2.17 K, flowing with zero viscosity, climbing the walls of its container and escaping. Helium-3 does not — until 0.0025 K, a thousand times colder, when its atoms pair up into effective bosons in the same way electrons do in a superconductor (Chapter 4.4). One neutron changes the transition temperature by a factor of a thousand.
Why you do not fall through the floor
This is where exclusion becomes something you can feel.
The naive answer — atoms are solid balls that touch — is wrong. An atom is almost entirely empty. If a nucleus were a marble, the nearest electron would be a kilometre away.
The next answer — electrostatic repulsion between electron clouds — is also wrong, or at least not the main effect. Ordinary matter is electrically neutral, and the attractive and repulsive terms very nearly cancel.
The real answer is exclusion. Push two atoms together and their electron clouds must overlap. But every low-energy state in the overlap region is already occupied. The incoming electrons are forced into higher energy states, and that costs energy. Resisting compression because compression costs energy is a force.
This is called degeneracy pressure, and it has nothing to do with charge. It would exist between neutral fermions with no electromagnetic interaction at all.
Estimate its size. For n electrons per unit volume, the Fermi energy — the energy of the highest occupied state — is:
E_F = \frac{\hbar^2}{2m_e}\left(3\pi^2n\right)^{2/3}
For a metal with n \approx 8.5\times10^{28} m⁻³ (copper, Chapter 4.4):
(3\pi^2n)^{2/3} = (2.52\times10^{30})^{2/3}
Taking logs: \log_{10}(2.52\times10^{30}) = 30.40, times 2/3 is 20.27, so the bracket is 1.86\times10^{20} m⁻².
E_F = \frac{(1.055\times10^{-34})^2}{2(9.109\times10^{-31})}\times1.86\times10^{20} = \frac{1.113\times10^{-68}}{1.822\times10^{-30}}\times1.86\times10^{20}
= (6.11\times10^{-39})(1.86\times10^{20}) = 1.14\times10^{-18}\ \text{J} = 7.1\ \text{eV}
Seven electron-volts, at absolute zero. Compare with k_BT at room temperature, 0.026 eV. The electrons in a metal are moving at enormous speeds not because they are hot but because they are forbidden from being slow — every low state is taken.
Their speed:
v_F = \sqrt{\frac{2E_F}{m_e}} = \sqrt{\frac{2(1.14\times10^{-18})}{9.109\times10^{-31}}} = 1.58\times10^{6}\ \text{m/s}
1600 km/s, which is the number Chapter 4.4 quoted and could not justify.
And the degeneracy pressure:
P = \frac{2}{5}nE_F = \frac{2}{5}(8.5\times10^{28})(1.14\times10^{-18}) = 3.9\times10^{10}\ \text{Pa}
Nearly 400,000 atmospheres, at room temperature, in a piece of copper. That is what makes solids hard.
Degeneracy pressure in stars
White dwarfs are held up entirely by electron degeneracy pressure, not by heat. A white dwarf has the Sun's mass in the Earth's volume, and it does not collapse because the electrons cannot be squeezed into fewer states.
There is a maximum mass. As the star gets heavier, the electrons are squeezed to relativistic speeds, and the pressure of a relativistic gas grows more slowly with density than a non-relativistic one. Above a critical mass, gravity wins.
M_{\text{Ch}} \approx 1.44\,M_\odot
The Chandrasekhar limit, computed by a 19-year-old Subrahmanyan Chandrasekhar in 1930 on the boat from India to England. Eddington, then the most influential astrophysicist alive, publicly ridiculed it, saying there ought to be a law of nature preventing a star behaving in so absurd a way. Chandrasekhar was right and received the Nobel Prize in 1983. Chapter 12.2 derives the limit.
Above it, the electrons are forced into protons, making neutrons, and the star becomes a neutron star held up by neutron degeneracy pressure — the same physics with a particle 1836 times heavier. Above about 2.2 solar masses even that fails, and nothing known can stop the collapse.
So the exclusion principle is what stops most of the mass in the universe from becoming black holes.
Bosons do the opposite
Bosons pile into the same state, and the consequences are equally dramatic.
Lasers (Chapter 5.6). Stimulated emission produces a photon identical to the trigger, and the reason it is stimulated — the reason the process is enhanced when many photons are already present — is that photons are bosons and the amplitude for entering an occupied state is larger.
Bose–Einstein condensation. Cool a gas of bosons enough and a macroscopic fraction of them occupies the single lowest state, all described by one wavefunction. Predicted by Bose and Einstein in 1924–25, achieved in 1995 with rubidium at 170 nanokelvin by Cornell and Wieman, and with sodium by Ketterle. All three shared the 2001 Nobel Prize.
Superfluidity. Liquid helium-4 below 2.17 K flows with exactly zero viscosity, passes through pores no ordinary liquid can enter, and climbs the walls of its container as a film to escape. A container of superfluid helium empties itself.
Superconductivity. Electrons are fermions and cannot condense — but below the critical temperature they pair into Cooper pairs, which have integer spin and are bosons, and those condense. That is the BCS theory mentioned in Chapter 4.4.
Adding angular momenta
Total angular momentum combines orbital and spin:
\vec{J} = \vec{L}+\vec{S}
with allowed magnitudes:
j = \ell+s, \ \ell+s-1,\ \dots,\ |\ell-s|
For one electron with s = \frac{1}{2}: if \ell = 0 then j = \frac{1}{2} only; if \ell = 1 then j = \frac{3}{2} or \frac{1}{2}.
And that splitting is fine structure. The two j values have slightly different energies because the electron's spin magnetic moment interacts with the magnetic field arising from its orbital motion. The sodium D lines at 589.0 and 589.6 nm are the 3p_{3/2} and 3p_{1/2} states decaying to 3s_{1/2} — one transition split into two by spin.
Spectroscopic notation, which appears throughout chemistry and atomic physics, is n\,^{2S+1}L_J. Sodium's ground state is 3\,^2S_{1/2}: n=3, one unpaired electron so 2S+1 = 2, L = 0 so the letter is S, and J = \frac{1}{2}.
Where this shows up in your life
Solidity itself. Every time you sit on a chair, you are being held up by the exclusion principle.
The periodic table and therefore all of chemistry, which follows from 2n^2 and the ordering of levels.
Every transistor and every LED, since the Fermi level and band filling are exclusion in a crystal.
MRI, which manipulates proton spins.
Superconducting magnets in MRI machines and the LHC, which need Cooper pairs.
Atomic clocks and GPS, which use hyperfine transitions between spin states.
And white dwarfs and neutron stars exist — meaning the heavy elements you are made of were dispersed by supernovae — because of a rule about wavefunction antisymmetry.
What the next chapter fixes
Spin gave two states from an intrinsic property. Now put two spins together, in a state where neither has a definite value alone but their relationship is fixed. Measure one and the other is instantly determined, however far apart they are. Einstein thought this proved quantum mechanics incomplete and published an argument to that effect in 1935. Thirty years later John Bell found a way to test who was right, and the experiments have now been done to overwhelming precision. Chapter 7.8 goes through the argument, the theorem and the results.