Quantum Statistics
Classical statistical mechanics counts states as though the particles carried labels, and the answer is wrong: the entropy is not extensive until a factor \(1/N!\) is inserted by hand (Statistical Mechanics), the heat capacities of gases and solids collapse at low temperature, and a metal's conduction electrons contribute almost nothing to its specific heat while contributing everything to its conductivity. This chapter replaces the counting. Identical particles are indistinguishable in principle, the states of \(N\) of them are either totally symmetric or totally antisymmetric under exchange, and the two cases give the Bose–Einstein and Fermi–Dirac distributions — from which the Planck spectrum, the degeneracy pressure that holds up a white dwarf, the linear electronic heat capacity of a metal, Pauli paramagnetism, Landau diamagnetism and Bose–Einstein condensation all follow. Each is stated below with the measurement that established it.
The chapter takes the exchange symmetry itself from Identical Particles and the spin–statistics theorem from Axiomatic Quantum Field Theory, and hands its results on to Experiment: Bose–Einstein Condensation (the atomic condensate as a laboratory system), Phonons and Lattice Dynamics and Electrons in Solids: Band Theory (solids), Superconductivity and Superfluidity (paired and condensed phases) and Compact Stars and Relativistic Astrophysics (degenerate stellar matter). Founding sources are Bose [Bose:1924], Einstein [Einstein:1924] [Einstein:1925], Fermi [Fermi:1926] and Dirac [Dirac:1926]; the standard modern treatments are [Landau:1980] [Pathria:2011].
Quantum Statistics: all derivations of this chapter are pending.
Indistinguishability
Counting states for identical particles
[Reserved: the difference between Boltzmann counting, in which exchanging two particles gives a new microstate, and the counting used by Bose, in which it does not [Bose:1924]; occupation-number representation and the specification of a state by \(\{n_{i}\}\) alone; the resulting combinatorics for unrestricted and restricted occupancy; the symmetrisation postulate and the exchange operator, developed in Identical Particles.]
The exclusion principle
[Reserved: Pauli's formulation — no two electrons in an atom share all four quantum numbers — introduced to explain the closing of electron shells and the anomalous Zeeman multiplet structure [Pauli:1925]; the periodic table and atomic spectra as the evidence base (Atomic Models and Spectra and Atoms and Molecules); the restatement as antisymmetry of the many-electron wavefunction and the Slater determinant.]
The electrons of a many-electron atom do not accumulate in the lowest orbital. The chemical periods close at atomic numbers \(Z=2,10,18,\dots\), the first ionization energy peaks at exactly those values and drops abruptly at the next, and the alkali spectra are those of a single valence electron outside an inert core: no two electrons of an atom share all four quantum numbers [Pauli:1925]. Nor is this a rule that softens at high precision. Passing a fresh current through a copper strip, so that the conduction electrons arriving at a given atom have never been there before, and searching the X-ray spectrum for a transition into an already-filled \(K\) shell, bounds the probability of a Pauli-violating state below about \(1.7\times10^{-26}\) [Ramberg:1990].
Exclusion derived rather than postulated: the symmetrisation of the state of identical particles, the vanishing of a totally antisymmetric state with two coincident single-particle labels, and the spin–statistics theorem that assigns antisymmetry to half-integer spin
The spin–statistics theorem
[Reserved: Pauli's proof that, in a relativistic quantum field theory with a positive-definite energy and local commutativity, half-integer spin fields must anticommute and integer spin fields must commute [Pauli:1940]; the failure of the alternative assignment — negative energies or negative norms — rather than mere inelegance; the axiomatic form of the theorem in Axiomatic Quantum Field Theory and its relation to the CPT theorem of Discrete Symmetries and CPT.]
Composite particles
[Reserved: Ehrenfest and Oppenheimer's rule that a bound system of an even number of fermions behaves as a boson and an odd number as a fermion, with the validity conditions — the exchange must not resolve the internal structure [Ehrenfest:1931]; the worked cases \(^{4}\mathrm{He}\) (boson) against \(^{3}\mathrm{He}\) (fermion), whose radically different low-temperature behaviour is the direct experimental demonstration; Cooper pairs and excitons as the same mechanism (Superconductivity and Superfluidity).]
Experimental limits on violations
[Reserved: the Ramberg–Snow experiment, in which a fresh current is passed through a copper strip and the X-ray spectrum searched for transitions to an already-filled \(K\) shell, bounding the probability of a Pauli-violating state below about \(1.7\times10^{-26}\) [Ramberg:1990]; later and stronger bounds from nuclear and electron stability; why an exact conservation law deserves a direct test rather than an assumption.]
The two distributions
Bose's derivation of the Planck law
[Reserved: Bose's counting of photon states in phase-space cells of volume \(h^{3}\), which derives the Planck spectrum without invoking classical electrodynamics at any point [Bose:1924]; Einstein's recognition and translation of the paper; the relation to the blackbody measurements of Black-Body Radiation and Planck's Hypothesis and to the cosmic microwave spectrum of Experiment: The Cosmic Microwave Background, the most precise blackbody ever measured; the photon chemical potential vanishing because photon number is not conserved.]
The Bose–Einstein distribution
[Reserved: Einstein's extension of Bose's counting to a gas of material particles, giving \(\langle n_{i}\rangle = [\ee^{\beta(\epsilon_{i}-\mu)}-1]^{-1}\) [Einstein:1924]; the constraint \(\mu \le \epsilon_{0}\); the enhancement of multiple occupancy relative to classical counting, which is the statistical statement behind stimulated emission (Quantum Optics and the Photon).]
Fermi–Dirac statistics
[Reserved: Fermi's quantisation of the ideal monatomic gas with the exclusion principle imposed [Fermi:1926] and Dirac's independent derivation from the antisymmetry of the wavefunction, with his identification of which particles obey which statistics [Dirac:1926]; the distribution \(\langle n_{i}\rangle = [\ee^{\beta(\epsilon_{i}-\mu)}+1]^{-1}\); the step function at \(T=0\) and the thermal smearing of width \(\sim kT\) about the chemical potential.]
The classical limit
[Reserved: both distributions reducing to \(\ee^{-\beta(\epsilon-\mu)}\) when the occupancies are small, i.e. when the thermal de Broglie wavelength is much less than the interparticle spacing; the degeneracy parameter \(n\lambda^{3}\) and its value for air at room temperature against a metal's conduction electrons; the recovery of the Sackur–Tetrode entropy of Statistical Mechanics with the \(1/N!\) now derived rather than inserted [Sackur:1911] [Tetrode:1912].]
Grand canonical treatment
[Reserved: why the grand canonical ensemble is the natural one — the occupation numbers are then independent and \(\log\Xi\) factorises over modes; the fugacity, the Bose and Fermi functions \(g_{\nu}(z)\) and \(f_{\nu}(z)\), and the resulting equations of state; the quantum virial expansion and the statistical “exchange pressure” that is repulsive for fermions and attractive for bosons in the absence of any interaction [Landau:1980].]
The ideal Bose gas
Einstein's condensation
[Reserved: the saturation of the excited-state population, so that below a critical temperature a macroscopic fraction of the particles must occupy the single-particle ground state [Einstein:1925]; Einstein's own statement that the phenomenon occurs without any interaction, purely from the statistics; the initial scepticism and the delay before it was taken to be physical.]
Critical temperature and condensate fraction
[Reserved: \(kT_{c} = 2\pi\hbar^{2}n^{2/3}/[m\,\zeta(3/2)^{2/3}]\) in three dimensions, the condensate fraction \(1-(T/T_{c})^{3/2}\), and the \(T^{3/2}\) heat capacity with its cusp at \(T_{c}\) [Einstein:1925] [Landau:1980]; the dimensional statement — no condensation of the ideal gas in a homogeneous system of fewer than three space dimensions — and the modification by a trapping potential; numerical values for helium and for a dilute alkali vapour.]
Superfluid helium-4
[Reserved: the abrupt loss of viscosity below \(2.17\,\mathrm{K}\) measured independently by Kapitza [Kapitza:1938] and by Allen and Misener [Allen:1938]; London's identification of the lambda transition with Bose–Einstein condensation, noting that the ideal-gas \(T_{c}\) at the density of liquid helium is \(3.1\,\mathrm{K}\), close to the observed \(2.17\,\mathrm{K}\) [London:1938]; the strong interactions that make the identification qualitative, the condensate fraction of only about \(10\,\mathrm{\%}\), and the full account in Superconductivity and Superfluidity.]
The weakly interacting Bose gas
[Reserved: Bogoliubov's diagonalisation of the weakly interacting Bose Hamiltonian, giving a phonon-like linear excitation spectrum at long wavelength and a depletion of the condensate [Bogoliubov:1947]; Landau's criterion for superfluidity, that a critical velocity exists whenever the excitation spectrum has no low-lying free-particle branch [Landau:1941]; why the ideal Bose gas, despite condensing, is not a superfluid.]
Dilute atomic condensates
[Reserved: the observation of condensation in rubidium-87 at \(170\,\mathrm{nK}\) by Anderson and coworkers [Anderson:1995] and in sodium at higher atom number by Davis and coworkers [Davis:1995]; the bimodal momentum distribution as the signature; laser and evaporative cooling, apparatus and uncertainties treated in Experiment: Bose–Einstein Condensation; the condensate as a nearly ideal realisation of Section 120.3.1, seventy years after the prediction.]
A gas of identical bosons cooled at fixed density undergoes a phase transition at a temperature set by the density and the mass alone, below which a macroscopic fraction of the atoms occupies one and the same single-particle state. The transition is observed in a dilute vapour of rubidium-87 at \(170\,\mathrm{nK}\) and in sodium at higher atom number [Anderson:1995] [Davis:1995], where the interactions are so weak that the ideal-gas prediction [Einstein:1925] applies with per-cent corrections; and it is observed in liquid helium-4, which loses its viscosity abruptly below \(2.17\,\mathrm{K}\) [Kapitza:1938] [Allen:1938] at a temperature close to the ideal-gas value of \(3.1\,\mathrm{K}\) computed at the density of the liquid [London:1938]. No attraction between the particles is involved: what condenses them is the counting rule.
Saturation of the excited-state population of the ideal Bose gas: that the sum over excited modes of the Bose occupation is bounded above at fixed temperature and volume when the chemical potential reaches the ground-state energy, so that any further particles must enter the ground state; the resulting critical temperature, condensate fraction and heat-capacity cusp in three dimensions
The ideal Fermi gas
Fermi energy and degeneracy pressure
[Reserved: the filled Fermi sphere at \(T=0\), the Fermi wavevector \(k_{F}=(3\pi^{2}n)^{1/3}\), the Fermi energy and Fermi temperature, and the ground-state pressure \(P = \tfrac{2}{5}n\epsilon_{F}\) which survives at zero temperature [Fermi:1926]; numerical values for the conduction electrons of copper (\(\epsilon_{F}\approx7\,\mathrm{eV}\), \(T_{F}\approx8\times 10^{4}\,\mathrm{K}\)), showing that a metal at room temperature is a strongly degenerate gas; the Fermi surface as an observable object.]
The Sommerfeld expansion
[Reserved: Sommerfeld's application of Fermi statistics to the electron theory of metals [Sommerfeld:1928], resolving the long-standing failure of the Drude model to explain why the electrons contribute a full Wiedemann–Franz conductivity but almost no heat capacity; the expansion in \(kT/\epsilon_{F}\) and the linear electronic heat capacity \(C = \gamma T\); the measured low-temperature form \(C = \gamma T + \beta T^{3}\) of a metal, whose two terms separate the electrons from the phonons of Phonons and Lattice Dynamics; band-structure corrections in Electrons in Solids: Band Theory.]
The conduction electrons of a metal carry the whole of its electrical and thermal current, as a free gas of one carrier per atom would, and yet contribute to its heat capacity about two orders of magnitude less than the \(\tfrac{3}{2}k_{B}\) per electron that classical equipartition demands. Between about \(1\,\mathrm{K}\) and \(5\,\mathrm{K}\) the heat capacity of a normal metal is measured to be
so that \(C/T\) plotted against \(T^{2}\) is a straight line whose intercept \(\gamma\) is electronic and whose slope \(\beta\) is the lattice contribution of Phonons and Lattice Dynamics; the separation was carried out for copper, silver and gold [Corak:1955]. A term linear in \(T\), and a coefficient this small, are what Fermi statistics predict and what Maxwell–Boltzmann statistics cannot produce [Sommerfeld:1928].
Derivation. Let \(g(\epsilon)\) be the density of one-electron states per unit volume, counting both spins. At temperature \(T\) the occupation \(f(\epsilon)=[\ee^{\beta(\epsilon-\mu)}+1]^{-1}\) differs from the zero-temperature step only within a shell of width of order \(k_{B}T\) about \(\mu\). Hence only about \(g(\epsilon_{F})k_{B}T\) electrons per unit volume are thermally active, and each carries an excess energy of order \(k_{B}T\), so the thermal energy density rises above its ground-state value by an amount of order \(g(\epsilon_{F})(k_{B}T)^{2}\), and the heat capacity is of order \(k_{B}^{2}Tg(\epsilon_{F})\) — linear in \(T\), not constant. The coefficient follows from the Sommerfeld expansion of \(\int H(\epsilon)f(\epsilon)\dd\epsilon\) in powers of \(k_{B}T/\mu\), whose leading correction is \((\pi^{2}/6)(k_{B}T)^{2}H'(\mu)\); applied to \(H=\epsilon g(\epsilon)\) and differentiated with respect to \(T\) it gives
which is Equation (120.1) with \(\gamma=\tfrac{1}{3}\pi^{2}k_{B}^{2}g(\epsilon_{F})\). For the free electron gas \(g(\epsilon_{F})=3n/(2\epsilon_{F})\) and Equation (120.2) becomes \(c_{V}=\tfrac{1}{2}\pi^{2}nk_{B}(T/T_{F})\) with \(k_{B}T_{F}=\epsilon_{F}\), smaller than the classical \(\tfrac{3}{2}nk_{B}\) by the factor \((\pi^{2}/3)(T/T_{F})\). With \(T_{F}\approx8\times 10^{4}\,\mathrm{K}\) for copper this factor is about \(10^{-2}\) at room temperature, which is the observed suppression. The measured \(\gamma\) is thus a direct measurement of the density of states at the Fermi level.
∎Pauli paramagnetism
[Reserved: Pauli's calculation of the spin susceptibility of a degenerate electron gas, temperature-independent and smaller than the Curie law by a factor \(\sim T/T_{F}\) [Pauli:1927a]; the resolution of the observed weak, temperature-independent paramagnetism of the alkali metals; the susceptibility as a measurement of the density of states at the Fermi level; the wider magnetism of Magnetism in Matter.]
The alkali metals are weakly paramagnetic with a susceptibility that is essentially independent of temperature, in flat contradiction with the Curie law \(\chi\propto1/T\) obeyed by the same number of localized moments, and smaller than the Curie value at room temperature by about two orders of magnitude [Pauli:1927a].
Derivation. In a field \(B\) the two spin populations acquire energies shifted by \(\mp\mu_{B}B\). They share one chemical potential, so the shift is absorbed by transferring electrons from the minority to the majority band until the two Fermi levels coincide; to first order in the field the transferred density is \(\tfrac{1}{2}g(\epsilon_{F})\mu_{B}B\) from each band, giving a magnetization
in which the temperature appears nowhere: only the electrons within \(k_{B}T\) of the Fermi surface can flip, and their number is proportional to \(T\) while the polarization each contributes is proportional to \(1/T\), so the two cancel. Dividing Equation (120.3) by the Curie susceptibility \(\chi_{\mathrm{C}}=n\mu_{0}\mu_{B}^{2}/k_{B}T\) of \(n\) free spins one half, and using \(g(\epsilon_{F})=3n/(2\epsilon_{F})\), gives \(\chi/\chi_{\mathrm{C}}=\tfrac{3}{2}(T/T_{F})\), of order \(10^{-2}\) at room temperature — the observed suppression, and the same degeneracy factor that suppresses the heat capacity in Phenomenon 120.3. The measured susceptibility is therefore a second, independent determination of \(g(\epsilon_{F})\).
∎Landau diamagnetism and quantum oscillations
[Reserved: Landau's demonstration that orbital quantisation in a magnetic field gives a diamagnetic contribution of exactly minus one third of the Pauli susceptibility for free electrons [Landau:1930]; the Landau levels and their degeneracy; the de Haas–van Alphen oscillations of the susceptibility periodic in \(1/B\), discovered in bismuth [deHaas:1930], which measure extremal Fermi-surface cross sections directly; the same levels underlying the quantum Hall measurements of Experiment: The Quantum Hall Effect.]
Degenerate Fermi gases in the laboratory
[Reserved: DeMarco and Jin's cooling of potassium-40 to \(T/T_{F}\approx 0.5\), with the onset of degeneracy seen as a deviation of the released energy from the classical value [DeMarco:1999]; the condensation of fermionic atom pairs across a Feshbach resonance and the crossover between a paired condensate and a Bardeen–Cooper–Schrieffer superfluid [Regal:2004]; the connection to Superconductivity and Superfluidity.]
Degenerate matter in astrophysics
White dwarfs
[Reserved: Fowler's identification of the electron degeneracy pressure of Section 120.4.1 as what supports a white dwarf against gravity, the first astrophysical application of the new statistics [Fowler:1926]; Chandrasekhar's relativistic treatment and the limiting mass near \(1.4\) solar masses, above which no degenerate configuration exists [Chandrasekhar:1931]; the observed white-dwarf mass distribution and the type Ia supernova connection, treated in Compact Stars and Relativistic Astrophysics and Stellar Structure and Nucleosynthesis.]
A white dwarf is a star of roughly solar mass compressed to roughly the radius of the Earth. It has no nuclear source, radiates freely, and is cooling — and yet it does not contract. The pressure that supports it does not vanish with the temperature, and is supplied by the exclusion principle acting on the electrons [Fowler:1926]. The support is not unlimited: since the electrons become relativistic as they are compressed, a degenerate configuration exists only below a limiting mass near \(1.4\) solar masses [Chandrasekhar:1931], and the observed white-dwarf masses respect it. The same argument applied to neutrons is bounded by the measured pulsar mass of \(1.97(4)\) solar masses, obtained from the Shapiro delay in a binary and sufficient by itself to exclude the softest nuclear equations of state [Demorest:2010].
The ground-state pressure of the ideal Fermi gas, its persistence at zero temperature, the softening of the equation of state when the Fermi momentum becomes relativistic, and the resulting limiting mass for a self-gravitating degenerate sphere
Neutron stars
[Reserved: neutron degeneracy pressure and the nuclear-matter corrections that dominate it, so that the maximum mass is set by the equation of state rather than by the ideal-gas result; the measurement of a pulsar of \(1.97(4)\) solar masses through the Shapiro delay, which excludes the softest equations of state [Demorest:2010]; the nuclear physics in Nuclear Forces and Nuclear Structure and the relativistic structure equations in Compact Stars and Relativistic Astrophysics.]
Correlations as direct evidence
Photon bunching
[Reserved: the Hanbury Brown and Twiss measurement of a positive correlation between the photocurrents of two detectors illuminated by the same thermal source [HanburyBrown:1956]; the initial disbelief and the resolution — the effect is a consequence of Bose statistics, not of any interaction; intensity interferometry as a technique for stellar angular diameters; the quantum-optical treatment of \(g^{(2)}(0)\) in Quantum Optics and the Photon.]
Bunching and antibunching compared
[Reserved: the direct side-by-side experiment of Jeltes and coworkers, measuring the two-particle correlation of clouds of metastable \(^{4}\mathrm{He}\) (bunching) and \(^{3}\mathrm{He}\) (antibunching) released from the same apparatus [Jeltes:2007]; the identical preparation of the two isotopes, which makes the opposite sign of the correlation an isolated demonstration of the statistics; the correlation length as the source coherence length.]
Identical particles arriving at two detectors from a chaotic source are correlated even though nothing couples them. Hanbury Brown and Twiss found the photocurrents of two detectors illuminated by one thermal light source to be positively correlated at small detector separation, an effect with no classical-particle explanation and none in any interaction [HanburyBrown:1956]. The sign of the correlation is fixed by the statistics alone, as the decisive control shows: clouds of metastable helium-4 and of helium-3, prepared and released from the same apparatus under the same conditions, give a two-particle correlation that is a peak for the boson and a dip for the fermion [Jeltes:2007].
The two-particle detection amplitude for identical particles from a chaotic source, the interference of the direct and exchanged terms, and the resulting second-order correlation of two at zero separation for bosons against zero for fermions, with the source coherence length as the correlation scale
Exchange effects in scattering
[Reserved: the interference of direct and exchange amplitudes in the scattering of identical particles, and Mott's cross section for two electrons with the resulting oscillatory term [Mott:1930]; the factor of two enhancement at \(90\,^\circ\) for identical bosons and the corresponding suppression for identical fermions; the measured alpha–alpha scattering pattern as the classic demonstration; scattering formalism in Scattering Theory.]
Beyond the two statistics
Why three space dimensions admit only two
[Reserved: the topology of the configuration space of identical particles in three space dimensions, whose exchange paths form the permutation group, so that only the two one-dimensional representations — symmetric and antisymmetric — are available [Leinaas:1977]; the spin–statistics theorem of Section 120.1.3 fixing which particles take which; the absence of any observed violation, per Section 120.1.5.]
Quasiparticle statistics in planar systems
[Reserved: electrons confined to a plane within a three-dimensional sample, for which the exchange paths form the braid group and the emergent quasiparticles may carry fractional exchange phases [Leinaas:1977]; the strict statement of scope — these are collective excitations of a physical three-dimensional material, not fundamental particles, and the treatise makes no claim about physics in fewer than three space dimensions; the fractional quantum Hall measurements that bear on it are reported in Experiment: The Quantum Hall Effect.]