Quantum Statistics: Fermi & Bose Gases
Quantum Statistics: Fermi & Bose Gases
Quantum indistinguishability profoundly changes statistical mechanics. Fermions (half-integer spin) obey the Pauli exclusion principle; bosons (integer spin) can crowd into the same state. These differences explain superconductivity, superfluidity, and lasers.
Definition
Fermi gas density of states in 3D: \(g(\epsilon) = (V/2\pi^2)(2m/\hbar^2)^{3/2}\sqrt{\epsilon}\). Total energy at \(T=0\): \(U_0 = \frac{3}{5}NE_F\). Heat capacity at low \(T\): \(C_V = \frac{\pi^2}{2}Nk_B(T/T_F)\).
Key Result
Bose-Einstein condensate forms below \(T_c = (2\pi\hbar^2/mk_B)(N/2.612V)^{2/3}\). Below \(T_c\), a macroscopic fraction occupies the ground state, giving rise to superfluidity.
Example 1
Electrons in copper (\(n = 8.5\times10^{28}\) m\(^{-3}\)): \(E_F = (\hbar^2/2m)(3\pi^2 n)^{2/3} \approx 7\) eV, \(T_F \approx 80{,}000\) K — far above room temperature, so copper electrons are deeply degenerate.
Example 2
Liquid helium-4 undergoes a superfluid transition at 2.17 K (the lambda point). Below this, helium flows without viscosity and exhibits quantized vortices — hallmarks of Bose-Einstein condensation.
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Practice
- Why does the specific heat of metals vary linearly with T at low temperatures?
- Explain Pauli paramagnetism of free electrons.
- What experimental signatures distinguish a Bose-Einstein condensate?
- How does the Fermi surface determine electrical conductivity?
Show Answer Key
1. At low $T$, only electrons near $\epsilon_F$ (within $\sim k_BT$) can be thermally excited. The number of excitable electrons $\propto k_BT/\epsilon_F$, each gaining $\sim k_BT$ energy. So $C_{\text{el}} = \frac{\pi^2}{2}Nk_B(T/T_F) \propto T$. Phonon contribution ($\propto T^3$ at low $T$) is negligible, so total $C \approx \gamma T$ at very low $T$.
2. Free electrons in a magnetic field: spin-up and spin-down bands split by $2\mu_B B$. More electrons fill the lower-energy spin-aligned band. Net magnetization $M = \mu_B^2 g(\epsilon_F) B$ (proportional to $B$ and density of states at Fermi level). This weak, temperature-independent (to leading order) paramagnetism contrasts with Curie paramagnetism ($\chi \propto 1/T$) of localized moments.
3. Experimental signatures: (1) Macroscopic quantum coherence — interference of BEC clouds. (2) Bimodal momentum distribution (narrow peak at $p=0$ superimposed on thermal distribution). (3) Anisotropic expansion (inverted aspect ratio after release from anisotropic trap). (4) Superfluid behavior (quantized vortices, reduced scattering).
4. The Fermi surface is the boundary in $k$-space between occupied and unoccupied states at $T=0$. Electrical conductivity depends on electrons near the Fermi surface (only they can be scattered to new states). Its shape determines anisotropy of conduction, de Haas–van Alphen oscillations, and nesting conditions for charge density waves. Measured via ARPES or quantum oscillation experiments.