Quantum Statistics and Bose–Einstein Condensate

Quantum Statistics: Two Types of Particles

In quantum mechanics, identical particles are indistinguishable—there is no “particle A” and “particle B”, there are simply “two electrons”. This fact has radical consequences for statistics. By the principle of indistinguishability: the wave function of a system of identical particles must be either symmetric (bosons) or antisymmetric (fermions) under the exchange of any two particles.

Fermions (half-integer spin: 1/2, 3/2, ...): Ψ(...i...j...) = −Ψ(...j...i...). Consequence (Pauli principle): two fermions cannot occupy the same quantum state.

Bosons (integer spin: 0, 1, 2, ...): Ψ(...i...j...) = +Ψ(...j...i...). Bosons “prefer” already occupied states.

Ideal Fermi Gas

Occupation numbers for fermions—Fermi–Dirac distribution:

⟨n_k⟩ = 1 / (e^{β(ε_k − μ)} + 1)

Symbols: ε_k—energy of level k; μ—chemical potential (determined from the condition of fixed N); β = 1/(k_BT).

At T = 0: ⟨n_k⟩ = 1 when ε_k < μ(0) ≡ E_F and 0 when ε_k > E_F. All levels below the Fermi level E_F are filled. Fermi energy:

E_F = (ℏ²/2m)(3π²n)^{2/3}

For electrons in copper: n ≈ 8.5 × 10²⁸ m⁻³, E_F ≈ 7 eV. Corresponding Fermi temperature T_F = E_F/k_B ≈ 81,000 K >> 300 K. Electrons in a metal at room temperature behave as a “cold” Fermi gas!

At T > 0: The distribution “blurs” near E_F over a band of width ~k_BT. Heat capacity: C_V^e = (π²/2) n k_B (T/T_F) << k_B. This explains why electrons contribute almost nothing to the heat capacity of metals at room temperature—a longstanding puzzle of classical physics!

Ideal Bose Gas and Bose–Einstein Condensate

Bose–Einstein distribution:

⟨n_k⟩ = 1 / (e^{β(ε_k − μ)} − 1)

Requirement: μ ≤ min(ε_k) = 0. As μ → 0⁻ with fixed N and T: if T is sufficiently low, the total number of particles in excited states “saturates”, and all “excess” particles collapse into the ground state ε₀ = 0. This is the Bose–Einstein Condensate (BEC).

Critical Temperature:

T_c = (2πℏ²/mk_B) · (n/ζ(3/2))^{2/3} ≈ 3.31 ℏ²n^{2/3}/(mk_B)

where ζ(3/2) ≈ 2.612 is the Riemann zeta function.

Condensate fraction: N₀/N = 1 − (T/T_c)^{3/2} for T < T_c

Physical meaning: For T < T_c, all atoms “collectively” enter a single quantum state with zero momentum—they form a single coherent quantum wave function. Analogy: laser light (all photons in one mode).

Numerical Example: Rubidium BEC

First observation of BEC (Cornell, Wieman, Ketterle—Nobel Prize 2001): ⁸⁷Rb at n ~ 10¹⁸ − 10¹⁹ m⁻³.

Mass of ⁸⁷Rb: m = 87 × 1.66 × 10⁻²⁷ kg ≈ 1.44 × 10⁻²⁵ kg.

At n = 10¹⁸ m⁻³: T_c = 3.31 × (1.055×10⁻³⁴)² × (10¹⁸)^{2/3} / (1.44×10⁻²⁵ × 1.38×10⁻²³) ≈ 170 nK!

This is a record for minimum temperatures—170 billion times colder than room temperature.

N₀/N at T = 0.5 T_c: 1 − (0.5)^{3/2} ≈ 1 − 0.354 ≈ 65% of atoms in the condensate.

Gross–Pitaevskii Equation

For an interacting BEC, the condensate wave function Ψ(r, t) satisfies:

iℏ ∂Ψ/∂t = [−ℏ²∇²/(2m) + V_ext + g|Ψ|²] Ψ

The nonlinear term g|Ψ|² describes interactions between atoms (g = 4πaℏ²/m, a—scattering length). This nonlinear Schrödinger equation predicts: condensate shape in the trap; dynamics—collective oscillations (sound); quantum vortices (analogue of vortices in a superfluid).

Connection with Superconductivity and Lasers

A superconductor is also a form of BEC: Cooper pairs (two electrons bound via a phonon)—bosons. At T < T_c they condense. The Gross–Pitaevskii equation for Cooper pairs → Ginzburg–Landau equation. The zero resistance is explained by the coherence of the condensate—scattering does not destroy the current.

A laser—stimulated emission into the same photon mode: also a variant of bosonic “condensate”. Laser coherence is analogous to BEC coherence.

Quantum Statistics in Superconductors and Atomic Technologies

Bose–Einstein and Fermi–Dirac quantum statistics are at the foundation of the most important modern technologies. Superconducting quantum interfaces used in quantum computers by IBM, Google, and IonQ operate at temperatures ~15 mK, where Cooper pairs bosonize and form a macroscopically condensed phase—a direct application of Bose–Einstein statistics. Electron transport in nanowires and quantum dots is described by the Fermi–Dirac distribution with level quantization, which is used to create single-electron transistors for ultrasensitive sensors. Dilution refrigerators—a technology for cooling to 10 mK—use Fermi statistics of helium isotopes (³He—fermion, ⁴He—boson). Atomic interferometers based on BEC condensates are used for geodesy (measuring free-fall acceleration with accuracy 10⁻¹⁰ g) and inertial navigation without GPS. A neutron star is a macroscopic object sustained by the pressure of the Fermi gas of neutrons against gravitational collapse: Fermi–Dirac statistics have an astrophysical scale.

Quantum degeneracy is not merely an academic interest. In a white dwarf’s plasma, electrons form a Fermi gas, where degeneracy pressure resists gravitational compression. The Chandrasekhar limit (1.4 solar masses) is the threshold at which Fermi pressure ceases to hold back gravity, determining the star’s fate: white dwarf or neutron star. Atomic interferometers with Bose condensate now surpass the best mechanical gravimeters in precision of measuring g by a factor of 100—a direct outcome of applying quantum statistics.

Assignment: (a) For ⁸⁷Rb at n = 10¹⁸ m⁻³: calculate T_c. At T = 100 nK: find N₀/N. (b) Electrons in a metal: for copper (n = 8.5×10²⁸ m⁻³) calculate E_F and T_F. Why does classical theory predict incorrect heat capacity? (c) What condenses in a superconductor? How do Cooper pairs differ from ordinary bosons?