Quantum Statistics in Detail

15.1 Fermi—Dirac and Bose—Einstein Distributions

For non-interacting quantum particles:

\langle n_i \rangle = \frac{1}{e^{\beta(\epsilon_i - \mu)} \pm 1}

Where $+$ is for fermions (Fermi—Dirac) and $-$ for bosons (Bose—Einstein).

Fermions (half-integer spin): Pauli exclusion limits $\langle n_i \rangle \leq 1$ .

Bosons (integer spin): No restriction on occupation number; $\langle n_i \rangle$ can diverge when $\epsilon_i = \mu$ .

15.2 The Fermi Gas

For a 3D gas of $N$ non-interacting fermions in volume $V$ :

N = \sum_{\mathbf{k}} \frac{1}{e^{\beta(\hbar^2 k^2/2m - \mu)} + 1} \xrightarrow{\text{continuum} \frac{V}{(2\pi)^3}\int d^3k\, f(\epsilon_k)}

The Fermi energy at $T = 0$ :

\epsilon_F = \frac{\hbar^2}{2m}(3\pi^2 n)^{2/3}

Where $n = N/V$ is the number density. The Fermi temperature is $T_F = \epsilon_F/k_B$ .

At low temperature ( $T \ll T_F$ ), the Sommerfeld expansion gives:

E = \frac{3}{5}N\epsilon_F\left[1 + \frac{5\pi^2}{12}\left(\frac{T}{T_F}\right)^2 + \cdots\right]

C_V = Nk_B\frac{\pi^2}{2}\frac{T}{T_F} + \cdots

The linear specific heat is a hallmark of degenerate Fermi systems.

15.3 The Bose Gas and Bose—Einstein Condensation

For bosons, the chemical potential must satisfy $\mu \leq \epsilon_0$ (ground state energy). When $\mu \to \epsilon_0$ A macroscopic fraction of particles condenses into the ground state.

The critical temperature for BEC in 3D:

T_c = \frac{2\pi\hbar^2}{mk_B}\left(\frac{n}{\zeta(3/2)}\right)^{2/3}

Where $\zeta(3/2) \approx 2.612$ .

Below $T_c$ The condensate fraction is:

\frac{N_0}{N} = 1 - \left(\frac{T}{T_c}\right)^{3/2}

Worked Example 15.1: Fermi Energy of Copper

Copper has one conduction electron per atom, atomic mass $63.5$ g/mol, density $8.96$ g/cm $^3$ .

n = \frac{8.96 \times 10^3 \text{ kg/m}^3}{63.5 \times 10^{-3} \text{ kg/mol} \times N_A = 1.41 \times 10^{29} \text{ m}^{-3} \times 6.022 \times 10^{23} = 8.49 \times 10^{28} \text{ m}^{-3}}

\epsilon_F = \frac{\hbar^2}{2m_e}(3\pi^2 \times 8.49 \times 10^{28})^{2/3}

= \frac{(1.055 \times 10^{-34})^2}{2 \times 9.109 \times 10^{-31}} \times (2.52 \times 10^{30})^{2/3}

= \frac{1.113 \times 10^{-68}}{1.822 \times 10^{-30}} \times 1.846 \times 10^{20}

= 6.11 \times 10^{-39} \times 1.846 \times 10^{20} = 1.13 \times 10^{-18} \text{ J}

T_F = \frac{1.13 \times 10^{-18}}{1.38 \times 10^{-23}} \approx 81\,900 \text{ K}

This is enormously higher than room temperature, confirming that conduction electrons in metals form a highly degenerate Fermi gas.

Worked Example 15.2: BEC in a Trap

For $N = 10^6$ rubidium-87 atoms in a harmonic trap with frequency $\bar{\omega}/(2\pi) = 100$ Hz:

In a harmonic trap, the density of states is $g(\epsilon) = \epsilon^2/(2\hbar^3\bar{\omega}^3)$ Giving:

T_c = \frac{\hbar\bar{\omega}}{k_B}\left(\frac{6N}{\pi^2\zeta(3)}\right)^{1/3}

= \frac{1.055 \times 10^{-34} \times 2\pi \times 100}{1.38 \times 10^{-23}}\left(\frac{6 \times 10^6}{\pi^2 \times 1.202}\right)^{1/3}

= \frac{6.63 \times 10^{-32}}{1.38 \times 10^{-23}} \times (5.07 \times 10^5)^{1/3}

= 4.81 \times 10^{-9} \times 79.7 \approx 383 \text{ nK}

This is consistent with the 1995 Cornell—Wieman BEC experiment.

Key Relationships

Concept	Relation	Significance
Distribution function	$\langle n_i \rangle = (e^{\beta(\epsilon_i-\mu)} \pm 1)^{-1}$	Unified form for FD/BE
Fermi energy	$\epsilon_F = \hbar^2(3\pi^2 n)^{2/3}/(2m)$	Sets scale for degenerate fermions
BEC critical temperature	$T_c = 2\pi\hbar^2 n^{2/3}/(mk_B \zeta(3/2)^{2/3})$	Phase transition temperature
Sommerfeld expansion	$C_V \propto T/T_F$	Linear specific heat at low $T$
Condensate fraction	$N_0/N = 1 - (T/T_c)^{3/2}$	Order parameter for BEC

Intuition

Quantum statistics is the story of two fundamentally different types of matter. Fermions, governed by the Pauli exclusion principle, refuse to share quantum states, creating a Fermi sea that fills up like a stadium seating chart. Bosons, by contrast, are social particles that pile into the same state, leading to Bose-Einstein condensation when cooled. At high temperatures both distributions merge into the classical Maxwell-Boltzmann form because quantum identity becomes irrelevant when particles are far apart. The Fermi energy sets the scale of electron behaviour in metals, while the BEC critical temperature marks the onset of macroscopic quantum coherence.

Common Pitfalls

Confusing FD and BE limits: At high temperature ( $T \gg T_F$ or $T \gg T_c$ ), both distributions reduce to the Maxwell—Boltzmann distribution. The quantum statistical corrections vanish when the interparticle spacing is much larger than the thermal de Broglie wavelength.
Chemical potential for bosons: For bosons, $\mu$ must always be less than the ground state energy. Setting $\mu > \epsilon_0$ gives negative occupation numbers, which is unphysical. At $T_c$ , $\mu \to 0$ (for a free gas with $\epsilon_0 = 0$ ).
Continuum approximation validity: The $g(\epsilon) \propto \sqrt{\epsilon}$ density of states assumes a large volume. For small systems (nanoparticles, quantum dots), the discrete level structure becomes important and the integral approximation fails.
BEC requires dimensionality: In 1D and 2D, Bose—Einstein condensation does not occur in a uniform gas (Hohenberg’s theorem). Trapping potentials can, however, create quasi-condensates in lower dimensions.

Applications

Metals and degenerate fermions: The electron specific heat $C_V = \gamma T$ in metals directly reflects the Fermi degeneracy. The Sommerfeld parameter $\gamma$ measures the density of states at the Fermi level.
Ultracold atoms: BEC in dilute atomic gases (Rb, Na, Li) enables studies of superfluidity, quantised vortices, and matter-wave interferometry.
Neutron stars: Degenerate neutron Fermi pressure supports neutron stars against gravitational collapse, with $\epsilon_F \sim 50$ MeV.
White dwarfs: Electron degeneracy pressure balances gravity, with the Chandrasekhar mass limit arising from relativistic Fermi gas physics.

Connections to Other Topics

Particle physics: Neutrinos in the early universe follow FD statistics. The neutrino background temperature ( $T_\nu \approx 1.95$ K) is slightly lower than the CMB due to $e^+e^-$ annihilation heating the photon bath.
Condensed matter: Heavy fermion materials have effective masses $m^* \gg m_e$ , producing Fermi temperatures $T_F \sim 10$ — $100$ K where quantum degeneracy meets accessible laboratory conditions.
Cosmology: The Bose—Einstein condensate has been proposed as a dark matter candidate (fuzzy dark matter), with a de Broglie wavelength of kiloparsec scales smoothing small-scale structure.
Quantum information: Degenerate Fermi gases in optical lattices simulate the Hubbard model, with the Pauli exclusion principle enforcing the no-double-occupancy constraint at half-filling.

Summary Table: Comparison of Quantum and Classical Gases

Property	Maxwell—Boltzmann	Fermi—Dirac	Bose—Einstein
Particles	Distinguishable	Indistinguishable fermions	Indistinguishable bosons
Occupation	$\langle n_i \rangle = e^{-(\epsilon_i-\mu)/k_BT}$	$\langle n_i \rangle \leq 1$	$\langle n_i \rangle$ unrestricted
Low- $T$ behaviour	All in ground state	Fermi sea	BEC
$C_V$ at low $T$	$\frac{3}{2}Nk_B$	$\propto T$	$\propto T^{3/2}$
Validity condition	$n\lambda_{\text{th}}^3 \ll 1$	$n\lambda_{\text{th}}^3 \gtrsim 1$	$n\lambda_{\text{th}}^3 \gtrsim 2.612$
Symmetry of wavefunction	No constraint	Antisymmetric	Symmetric

Cross-References

Classical Limit and the Maxwell-Boltzmann Distribution: The Maxwell-Boltzmann distribution emerges as the high-temperature limit of both quantum distributions derived here.
Bose-Einstein Condensation: BEC is the macroscopic occupation of the ground state predicted by Bose-Einstein statistics at low temperatures.
Fermi Gas at Finite Temperature: The Fermi-Dirac distribution is applied to electrons in metals, giving the linear specific heat and Fermi surface.
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