Identical Particles
Nothing in the postulates of The Postulates of Quantum Mechanics yet says what happens when two particles of a system cannot be told apart. Classically the question barely arises: two billiard balls may agree in every measurable property and still be labelled, because each carries its own trajectory. Quantum mechanically there is no trajectory to carry a label, the labels are therefore unobservable, and the demand that no prediction depend on them turns out to be among the strongest constraints in physics. It splits matter into exactly two families, forbids two electrons from occupying the same state [Pauli:1925], supplies the effective force that orders magnets [Heisenberg:1926] [Dirac:1926], fixes the size of atoms and the stability of bulk matter [Dyson:1967] [Lieb:1975], and holds white dwarfs up against their own gravity [Chandrasekhar:1931]. Its classical shadow is already visible in the Gibbs paradox of Statistical Mechanics [Gibbs:1902].
The chapter sits after the general apparatus of The Postulates of Quantum Mechanics and Angular Momentum and Spin and Approximation Methods because it needs all three: a Hilbert space to act on with permutations, spin to build the two families out of, and perturbation theory to turn the exchange term into a number. It is the last piece of one-particle quantum mechanics and the first piece of every many-body theory that follows — the multi-electron atoms and molecules of Atoms and Molecules, the electron gas of Electrons in Solids: Band Theory, the quantum statistics of Quantum Statistics, and the condensates of Experiment: Bose–Einstein Condensation. Helium is the worked example throughout. The deepest statement of the chapter, the spin–statistics connection [Pauli:1940], is stated here and proved only in Axiomatic Quantum Field Theory, since its hypotheses are relativistic and field-theoretic and are not available at this point in the book. Standard treatments are [Sakurai:2017] [CohenTannoudji:1977].
Identical Particles: all derivations of this chapter are pending.
Indistinguishability and the exchange operator
What indistinguishability asserts
[Reserved: the physical assertion — that no measurement can distinguish particle \(1\) from particle \(2\), so that every observable \(A\) of an \(N\)-particle system commutes with every permutation of the labels; the contrast with the classical case, where identical particles are still individuated by their trajectories, and the resulting overcounting that Gibbs identified in the entropy of mixing [Gibbs:1902] and that the \(1/N!\) of Statistical Mechanics patches by hand; the statement that “identical” means agreement in every intrinsic quantum number — mass, charge, spin, lepton and baryon number — and the experimental fact that electrons agree in all of them to the precision of Experiment: The Electron Anomalous Magnetic Moment.]
The permutation group on the Hilbert space
[Reserved: the symmetric group \(S_{N}\) acting on \(\mathcal{H}^{\otimes N}\) by permuting tensor factors; that this is a unitary representation, reducible in general, whose irreducible components are labelled by Young diagrams; the group-theoretic language being that of Linear Algebra and Representation Theory; the observation that the totally symmetric and totally antisymmetric one-dimensional representations are distinguished among all of them by being the only ones in which every transposition acts by a single number.]
The exchange operator and its eigenvalues
[Reserved: the transposition operator \(P_{12}\), its unitarity and its involutive property \(P_{12}^{2}=\identity\), hence eigenvalues \(\pm1\); that \(\comm{P_{12}}{\Ham}=0\) for a Hamiltonian symmetric under exchange, so exchange parity is a constant of the motion and a state cannot change family under any evolution; the resulting superselection rule, first formulated as an independent postulate by Messiah and Greenberg [Messiah:1964].]
The symmetrization postulate
[Reserved: the postulate that the physical states of \(N\) identical particles span only the totally symmetric or the totally antisymmetric subspace of \(\mathcal{H}^{\otimes N}\), never a mixed-symmetry one; its statement in Dirac's 1926 paper [Dirac:1926] and in Heisenberg's resonance paper [Heisenberg:1926]; the careful modern formulation, which separates the postulate from the spin–statistics theorem that later explains it [Messiah:1964]; that this is a restriction on the state space, not a restriction on the dynamics, and that it is logically independent of everything in The Postulates of Quantum Mechanics.]
Bosons and fermions
Bose–Einstein statistics
[Reserved: the symmetric sector; Bose's counting of photon states [Bose:1924] and Einstein's extension to a material ideal gas, with the prediction of condensation below a critical temperature [Einstein:1925]; the enhancement factor \(\sqrt{n+1}\) for adding a particle to an occupied mode, which is the origin of stimulated emission (Quantum Optics and the Photon); pointers to the thermodynamics in Quantum Statistics and to the observation of condensation in Experiment: Bose–Einstein Condensation.]
Fermi–Dirac statistics and the exclusion principle
[Reserved: the antisymmetric sector; Pauli's exclusion principle, announced as an empirical rule that closes atomic shells and explains the periodic table [Pauli:1925], and its restatement as antisymmetry of the wavefunction; the ideal gas obeying it, worked out independently by Fermi [Fermi:1926] and Dirac [Dirac:1926]; the Fermi sea, Fermi energy and Fermi surface as they will be used in Electrons in Solids: Band Theory and Semiconductors; the statement that exclusion is a property of the state space and not a force — nothing pushes.]
The chemical elements repeat their properties in periods of \(2,8,8,18,18,32\); the ionization energy climbs through each period and falls abruptly at the start of the next; and the elements ending each period are chemically inert. Pauli extracted from the spectroscopic evidence the rule behind it: no two electrons of an atom occupy states agreeing in all four quantum numbers, so that a shell of principal quantum number \(n\) holds at most \(2n^{2}\) electrons [Pauli:1925]. Nothing in The Postulates of Quantum Mechanics forbids the alternative. Were the electrons free to share the lowest orbital, every atom would have the ground configuration \(1s^{N}\), all elements would be chemically alike, and there would be no chemistry and no periodic table to explain.
Derivation. Take the exclusion rule as stated — at most one electron per one-electron state — and count the states. In a central field a one-electron state is labelled by \(n\), by \(\ell=0,1,\ldots,n-1\), by \(m_{\ell}=-\ell,\ldots,\ell\) and by the two values of \(m_{s}\), the ranges of \(\ell\) and \(m_{\ell}\) being those established in Angular Momentum and Spin. The number of states of a given \(n\) is therefore
which is \(2,8,18,32\) for \(n=1,2,3,4\): the observed period lengths, and the observed inert gases are the elements at which a group of these states is exactly filled. The qualification the data impose is that the \(\ell\) degeneracy of The Hydrogen Atom is broken in a many-electron atom, so the subshells fill out of order — \(4s\) before \(3d\) — and a period of \(18\) is first entered in the fourth row; and the periods of \(8\) are the \(s\) and \(p\) subshells of one \(n\), which hold \(2+6\) between them. What Equation (83.1) does not contain is a force. No term in the Hamiltonian was modified; the counting restricts which states exist, and that restriction alone produces the periodic table.
∎Parastatistics, and why nature does not use it
[Reserved: the mixed-symmetry representations of \(S_{N}\) are not excluded by any principle of The Postulates of Quantum Mechanics, and Green's generalized field quantization [Green:1953] realizes them consistently as parabosons and parafermions of order \(p\); the observational situation is that every known particle has \(p=1\); the theoretical statement that in a local relativistic field theory parastatistics of order \(p\) is equivalent to ordinary statistics with a hidden \(\SU(p)\) degree of freedom, which is how colour entered Quantum Chromodynamics; the experimental bounds are those of Section 83.5.2.]
Exchange in an effectively two-dimensional system
[Reserved: the argument that only two families exist uses the topology of the configuration space of \(N\) points, and in three space dimensions the exchange paths fall into two homotopy classes; Leinaas and Myrheim showed that for particles confined to a plane the relevant group is the braid group and a continuum of phases is allowed [Leinaas:1977], with Wilczek naming the resulting excitations anyons [Wilczek:1982]. This treatise instantiates \(3+1\) dimensions only, so the statement belongs here as a property of an effectively two-dimensional system inside ordinary space — a two-dimensional electron gas in a strong magnetic field — and not as a statement about spacetime. The evidence is the fractional quantum Hall effect of Experiment: The Quantum Hall Effect and the direct anyon-collision measurement of [Bartolomei:2020].]
Constructing many-particle states
Slater determinants and permanents
[Reserved: the antisymmetrizer applied to a product of \(N\) orthonormal orbitals, giving the determinant of the matrix \(\psi_{i}(x_{j})\) normalized by \(1/\sqrt{N!}\), introduced by Slater in his theory of complex spectra [Slater:1929]; that it vanishes identically when two orbitals coincide, which is the exclusion principle read off an algebraic identity; the bosonic counterpart, the permanent, and why it is computationally hard where the determinant is easy; matrix elements of one- and two-body operators between determinants — Slater's rules — as the working tool of Atoms and Molecules.]
Occupation numbers and second quantization
[Reserved: the passage from labelled particles to occupation numbers of modes; Fock space and its construction [Fock:1932]; creation and annihilation operators with the commutator algebra of Jordan and Klein for bosons [Jordan:1927] and the anti commutator algebra of Jordan and Wigner for fermions [Jordan:1928], the latter written down precisely to make the Pauli exclusion principle an operator identity; that symmetrization is then automatic rather than imposed; the observation that this formalism is already field theory and is taken up as such in Canonical Quantization of Fields.]
The self-consistent field
[Reserved: the variational determination of the best single Slater determinant; Hartree's self-consistent field [Hartree:1928] and Fock's antisymmetrized correction [Fock:1930], giving the Hartree–Fock equations with their direct and exchange terms; the definition of correlation energy as what the method misses; helium and the light atoms as the test cases, and the precision benchmark of Hylleraas' explicitly correlated calculation [Hylleraas:1929]; forward pointer to Atoms and Molecules for atoms and molecules and to Electrons in Solids: Band Theory for the extended case.]
The exchange interaction
Exchange energy has no classical analogue
[Reserved: the two-electron matrix element of the Coulomb repulsion between symmetrized and antisymmetrized states splits into a direct term \(J\) and an exchange term \(K\), with the sign of \(K\) entering with the exchange parity; Heisenberg's resonance treatment [Heisenberg:1926] and Dirac's derivation of the same splitting from antisymmetry [Dirac:1926]; the essential point that no new interaction has been introduced — the exchange term is the ordinary electrostatic interaction evaluated on a constrained state space, which is why it is large, of order electronvolts, while the magnetic dipole–dipole energy between the same two spins is smaller by some three orders of magnitude.]
Helium: ortho and para
[Reserved: the worked example of the chapter. The two-electron state factorizes into a spatial and a spin part whose exchange parities must multiply to \(-1\): the spin singlet takes the symmetric spatial function (parahelium) and the spin triplet the antisymmetric one (orthohelium). Consequences to be derived: the absence of a triplet ground state; the \(2\,^{3}\!S\)–\(2\,^{1}\!S\) splitting of about \(0.80\,\mathrm{eV}\), entirely exchange in origin; the near-absence of intercombination lines, and hence the nineteenth-century appearance of helium as two distinct gases; the metastability of \(2\,^{3}\!S\). Perturbative treatment via Approximation Methods, variational benchmark via [Hylleraas:1929], and modern spectroscopic precision via Experiment: Precision Spectroscopy and Atomic Clocks.]
The spectrum of helium falls into two nearly independent systems, singlet and triplet, with transitions between them so weak that the nineteenth century took them for the spectra of two different gases. The lowest term of the entire spectrum is a singlet: no triplet state of the \(1s^{2}\) configuration exists at all, though the corresponding singlet does. Where both do exist — the \(1s2s\) configuration — the triplet lies lower, by about \(0.80\,\mathrm{eV}\). That separation cannot be magnetic: the magnetic interaction of two electron moments a tenth of a nanometre apart is of order \(5\times 10^{-5}\,\mathrm{eV}\), four orders of magnitude too small. Heisenberg and Dirac identified it instead as an ordinary electrostatic energy, evaluated on states of definite exchange symmetry [Heisenberg:1926] [Dirac:1926].
Derivation. Electrons are fermions, so the two-particle state changes sign under the exchange of the pair of labels carrying both position and spin. Neglecting spin-dependent forces, the Hamiltonian does not couple space to spin and a state may be taken as a product \(\Psi=\psi(\vect{r}_{1},\vect{r}_{2})\,\chi(1,2)\) of a spatial and a spin factor, each of definite exchange parity. Antisymmetry of the product then requires the two parities to be opposite:
For the configuration \(1s^{2}\) both electrons occupy the same spatial orbital, so \(\psi=\varphi_{1s}(\vect{r}_{1})\,\varphi_{1s}(\vect{r}_{2})\) is necessarily symmetric, and by Equation (83.2) the spin state must be the antisymmetric singlet. A triplet ground state is not merely higher in energy; there is no such state to have an energy.
The sign of the \(1s2s\) splitting follows from the same parity. An antisymmetric \(\psi\) vanishes wherever \(\vect{r}_{1}=\vect{r}_{2}\) and so keeps the two electrons apart, while a symmetric one does not — indeed it is enhanced there. Evaluating the Coulomb repulsion \(e^{2}/4\pi\varepsilon_{0}\abs{\vect{r}_{1}-\vect{r}_{2}}\), a positive operator, on the two states therefore costs more in the symmetric case, so the singlet lies above the triplet, as observed. No spin operator appeared anywhere in the argument: the spin merely selected which spatial function was admissible, and the energy difference is electrostatic. That is why it is of the order of an electronvolt while the magnetic interaction between the same two moments is smaller by four orders of magnitude, and it is the mechanism carried over to magnetic order in Section 83.4.3.
∎From exchange to magnetic order
[Reserved: Dirac's replacement of the exchange term by an effective spin–spin coupling \(-2K\,\vect{S}_{1}\cdot\vect{S}_{2}\) [Dirac:1926], and Heisenberg's use of it to explain ferromagnetism as a cooperative alignment driven by electrostatics [Heisenberg:1928]; the sign of \(K\) selecting ferromagnetic or antiferromagnetic order; that the Curie temperatures of iron, cobalt and nickel are of the electronvolt scale precisely because the mechanism is Coulomb and not magnetic; Hund's first rule as the atomic case; full development in Magnetism in Matter, and the covalent bond as the same computation with the opposite sign convention in Atoms and Molecules.]
The spin–statistics connection
Statement of the theorem
[Reserved: the empirical regularity that half-integer-spin particles are fermions and integer-spin particles are bosons, and its elevation to a theorem by Pauli [Pauli:1940] on the hypotheses of relativistic invariance, positivity of the energy and locality (the vanishing of commutators of observables at spacelike separation); the sharpened axiomatic version of Lüders and Zumino [Lueders:1958]. The theorem is stated here and proved in Axiomatic Quantum Field Theory, because none of its hypotheses is available in nonrelativistic quantum mechanics — which is itself the finding: the symmetrization postulate of Section 83.1.4 is a genuine extra postulate at this level of the theory and stops being one only in a relativistic field theory. Its companion, the CPT theorem, is in Discrete Symmetries and CPT. Composite particles inherit the statistics of the parity of their constituent fermion number, which is why \(^{4}\)He condenses and \(^{3}\)He must pair before it can (Superconductivity and Superfluidity).]
Experimental tests of the symmetrization postulate
[Reserved: the postulate is testable and has been tested. The Ramberg–Snow experiment passed a large current through a copper strip and searched for the X-ray line that a new electron making a Pauli-forbidden transition to a filled \(K\) shell would emit, bounding the violation probability at the level of \(10^{-26}\) [Ramberg:1990]; the underground VIP-2 continuation improves it [Napolitano:2022]. Complementary bounds come from the non-observation of symmetric two-photon states and from nuclear spectroscopy. The section will tabulate the quantity bounded, the bound, and the assumption each experiment tests, in the sense of Epistemology and the Scientific Method: the point is that “electrons are identical” is a measured statement, not a definition.]
The symmetrization postulate is a testable statement about the world and it has been tested directly. Ramberg and Snow passed a large current through a thin copper strip and searched its X-ray spectrum for the line that a newly arrived electron would emit on making a transition into an already filled \(K\) shell — a transition forbidden by exclusion, and displaced in energy from the ordinary \(K\alpha\) line by the extra screening that the doubly occupied shell provides. No such line appeared above background, which bounds the probability that a fresh electron enters a filled state at the level of \(10^{-26}\) [Ramberg:1990]. The VIP-2 apparatus, the same experiment moved underground so that cosmic rays no longer set the background, has improved the limit [Napolitano:2022]. That “electrons are identical” is therefore an experimental result carrying a number, and not a convention of the formalism.
The Ramberg–Snow bound: the energy displacement of the anomalous X-ray line produced by the extra screening of a doubly occupied inner shell, the number of fresh electrons that a given charge delivers into the target lattice, and the conversion of a null count and its background into an upper limit on the violation probability
Two-particle interference
The Hong–Ou–Mandel dip
[Reserved: two indistinguishable photons entering the two input ports of a balanced beam splitter always leave together, because the two amplitudes for coincident detection cancel for bosons; the measurement of the resulting coincidence dip by Hong, Ou and Mandel [Hong:1987], whose width measures the photon wave-packet duration on a subpicosecond scale and whose depth measures indistinguishability directly. The derivation is a two-line calculation with the bosonic commutators of Section 83.3.2 and belongs inline. Uses: it is the standard indistinguishability metric in Quantum Optics and the Photon, and the entanglement-swapping step of Entanglement and Bell Tests depends on it.]
Send one photon into each of the two input ports of a balanced beam splitter. If the two are made indistinguishable — same polarization, same spectrum, arrival times matched — the detectors at the two output ports never fire together: the coincidence rate falls to zero. Delaying one photon by more than its wave-packet duration restores the coincidences to the value expected of two independent particles, so the coincidence rate plotted against delay shows a dip. Hong, Ou and Mandel measured that dip and read the photon wave-packet duration off its width, on a subpicosecond scale that no detector of theirs could have resolved directly [Hong:1987]. Each photon separately divides evenly between the ports; only the pair refuses to.
Derivation. Let \(a^{\dagger}\) and \(b^{\dagger}\) create a photon in the two input modes and \(c^{\dagger}\), \(d^{\dagger}\) in the two output modes. A lossless balanced beam splitter acts on the creation operators as
the relative minus sign being forced by unitarity: without it the map would not preserve the commutators. The input with one photon in each port is \(a^{\dagger}b^{\dagger}\ket{0}\), so
For bosons \(\comm{c^{\dagger}}{d^{\dagger}}=0\), so the last two terms of Equation (83.3) cancel identically and, since \(c^{\dagger}c^{\dagger}\ket{0}=\sqrt{2}\ket{2_{c},0_{d}}\),
Every surviving term of Equation (83.4) has both photons in one port: the amplitude for one photon in each output is exactly zero, which is the observed vanishing of the coincidence rate. The cancellation is between the two histories that end in the same final state — both photons reflected, both transmitted — and it survives only while nothing distinguishes them, so anything that labels the photons, a difference in arrival time included, fills the dip in. Its depth is therefore a direct measurement of indistinguishability.
Repeating the calculation for fermions changes one thing and reverses the conclusion. The anticommutator \(\acomm{c^{\dagger}}{d^{\dagger}}=0\) makes the cross terms add rather than cancel, while \(c^{\dagger}c^{\dagger}=d^{\dagger}d^{\dagger}=0\) kills the terms that survived above, leaving \(-c^{\dagger}d^{\dagger}\ket{0}\): the two particles leave in opposite ports with certainty. Exchange parity is read directly off the sign of a two-particle coincidence rate.
∎Intensity correlations and the fermionic counterpart
[Reserved: the Hanbury Brown–Twiss observation that photons from a thermal source arrive in correlated pairs [HanburyBrown:1956], its initial reception as a paradox, and its resolution as bosonic bunching — an interference of two-particle amplitudes with no one-particle analogue; the corresponding antibunching of electrons measured in a mesoscopic beam splitter [Henny:1999]; the sign of the correlation as a direct readout of exchange parity, and its use as a source-size measurement in stellar intensity interferometry and in heavy-ion collisions.]
Photons from a thermal source do not arrive independently of one another. Splitting the light of a source between two photodetectors and correlating their output currents, Hanbury Brown and Twiss found an excess of coincidences at zero delay over the rate that statistically independent arrivals would give, decaying away over the coherence time of the light [HanburyBrown:1956]. The result was disputed on the ground that photons detected at separate places should know nothing of each other, and it is nevertheless there. The same measurement made on electrons leaving a mesoscopic beam splitter gives the opposite sign, a deficit of coincidences [Henny:1999]. The correlation is a two-particle effect with no one-particle counterpart: neither detector alone shows anything unusual, and the sign of the correlation between them reports the exchange parity of the particles directly.
The two-particle correlation of a chaotic source: the sum over the two indistinguishable histories by which one particle from each of two source points reaches the two detectors, the factor of two it produces in the zero-delay coincidence rate for symmetric states, and its replacement by zero for antisymmetric ones, together with the loss of the effect once the path difference exceeds the coherence length
Consequences for bulk matter
Stability of matter
[Reserved: that the ground-state energy of \(N\) electrons and \(N\) nuclei is bounded below by a constant times \(N\) — so that bulk matter neither collapses nor releases unbounded energy on assembly — is a theorem, and it is false without the exclusion principle. Dyson and Lenard proved it first [Dyson:1967]; Lieb and Thirring gave the proof that yields a realistic constant, via their kinetic-energy inequality [Lieb:1975]. The section states the theorem, records that bosonic “matter” would instead have energy growing like \(-N^{7/5}\), and refers the proof to Appendix A. This is the sharpest answer available to the question of why atoms have a size and why matter is extended and nearly incompressible.]
Degeneracy pressure
[Reserved: the pressure of a degenerate Fermi gas at zero temperature, derived from the exclusion principle alone; its nonrelativistic and ultrarelativistic forms and the resulting Chandrasekhar limit of about \(1.4\) solar masses for a white dwarf [Chandrasekhar:1931]; the neutron-star analogue; the same mechanism, at laboratory densities, as the incompressibility of solids and the bulk modulus of a metal. Astrophysical development in Compact Stars and Relativistic Astrophysics and Stellar Structure and Nucleosynthesis, thermodynamic development in Quantum Statistics.]
There exist stars whose mass is of the order of the Sun's and whose radius is of the order of the Earth's, so that their mean density exceeds that of ordinary matter by some six orders of magnitude. They have exhausted their nuclear fuel, they radiate, and they do not contract; and as they cool they still do not contract, so whatever supports them is not thermal pressure. What supports them is the pressure of a degenerate electron gas, which is a consequence of the exclusion principle alone and does not vanish at zero temperature. There is also a ceiling. No white dwarf is observed above about \(1.4\) times the mass of the Sun, and Chandrasekhar showed that none can exist, because at the densities a larger mass would require the electrons move relativistically and the equation of state softens until it can no longer hold the star up [Chandrasekhar:1931].
The degenerate electron gas: its zero-temperature pressure obtained from the exclusion principle and the filling of momentum space up to the Fermi momentum, the change of the exponent relating pressure to density once that momentum exceeds the electron mass times the speed of light, and the resulting limiting mass, which is of the order of the cube of the Planck mass divided by the square of the nucleon mass