Quantum Statistics

Identical quantum particles obey different statistics than classical particles. Fermions (half-integer spin) obey the Pauli exclusion principle, leading to Fermi-Dirac statistics. Bosons (integer spin) can pile into the same state, leading to Bose-Einstein statistics.

Key Concepts

Fermions obey Pauli exclusion: at most 1 per state
Fermi-Dirac: n_FD = 1/(e^{(ε-μ)/k_BT} + 1)
Bosons can have any occupation: n_BE = 1/(e^{(ε-μ)/k_BT} - 1)
Fermi energy E_F: all states below E_F filled at T=0
Bose-Einstein condensation below critical temperature T_c

Key Equations

Fermi-Dirac distribution

n_{FD}(\varepsilon) = \frac{1}{e^{(\varepsilon-\mu)/k_BT}+1}

Bose-Einstein distribution

n_{BE}(\varepsilon) = \frac{1}{e^{(\varepsilon-\mu)/k_BT}-1}

Fermi energy (3D free electrons)

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

BEC critical temperature

T_c = \frac{2\pi\hbar^2}{mk_B}\left(\frac{n}{2.612}\right)^{2/3}

Worked Example

Example Problem

Problem

Find the Fermi-Dirac occupation at ε = E_F at any T.

Solution

n_FD(E_F) = 1/(e^0 + 1) = 1/2. The chemical potential μ = E_F at T=0 and ≈E_F at low T; at ε=μ, n_FD=1/2 always.

Practice

Exercises

7 problems

1 of 7

Find the Fermi-Dirac occupation n_FD for a state at ε = μ + k_BT.

Your answer

2 of 7

Find n_BE for a boson state at ε = μ + k_BT (with μ < ε, so argument is +1).

Your answer

3 of 7

Copper has electron density n=8.49×10²⁸ m⁻³. Find the Fermi energy E_F in eV. (m=9.11×10⁻³¹ kg)

Unlock Exercise 3