Quantum Statistics in Detail | Physics

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 $\epsilon_i = \mu$.

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.

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:

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

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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.

1. 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.

2. 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 $\epsilon_0 = 0$).

3. 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.

4. 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.

• 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.

• 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.

• 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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This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Ensure you have mastered the prerequisite material before attempting this advanced content.

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Ensure you have mastered the prerequisite material before attempting this advanced content.