← Part VII/Fermions & Bosons

2. Fermions & Bosons

Reading time: ~28 minutes | Pages: 7

Two fundamental classes of particles with profoundly different behaviors.

Classification by Spin

Fermions (half-integer spin):

Spin $s = 1/2, 3/2, 5/2, \ldots$
Obey Fermi-Dirac statistics
Antisymmetric wave functions
Examples: electrons, protons, neutrons, quarks, neutrinos

Bosons (integer spin):

Spin $s = 0, 1, 2, \ldots$
Obey Bose-Einstein statistics
Symmetric wave functions
Examples: photons, gluons, W/Z bosons, Higgs boson, $^4$He atoms

Occupation Numbers

Fermions:

$$n_i = 0 \text{ or } 1$$

Each quantum state can contain at most one fermion

Bosons:

$$n_i = 0, 1, 2, 3, \ldots, \infty$$

Unlimited number of bosons can occupy same state

Distribution Functions

Fermi-Dirac distribution:

$$f_{FD}(E) = \frac{1}{e^{(E-\mu)/k_BT} + 1}$$

Average occupation number at energy $E$, chemical potential $\mu$

Bose-Einstein distribution:

$$f_{BE}(E) = \frac{1}{e^{(E-\mu)/k_BT} - 1}$$

Classical limit ($T \to \infty$): Both approach Maxwell-Boltzmann

$$f_{MB}(E) \approx e^{-(E-\mu)/k_BT}$$

Fermi Energy and Fermi Surface

At $T = 0$, fermions fill states up to Fermi energy:

$$E_F = \mu(T=0)$$

For free electron gas in 3D:

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

where $n$ is number density

Fermi surface: Boundary in momentum space between occupied and unoccupied states

Degeneracy Pressure

Fermions resist compression due to Pauli exclusion:

$$P_{deg} = \frac{2}{5}nE_F$$

Applications:

White dwarfs: Electron degeneracy pressure supports star against gravity
Neutron stars: Neutron degeneracy pressure (even stronger)
Metals: Electrons near Fermi surface determine properties

Bose-Einstein Condensation

Below critical temperature, macroscopic occupation of ground state:

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

Fraction in ground state:

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

Observed in:

Liquid $^4$He (superfluid)
Ultracold atomic gases (87Rb, 23Na, etc.)
Exciton-polaritons in semiconductors

Photons and Blackbody Radiation

Photons are bosons with $\mu = 0$ (not conserved):

$$f(\omega) = \frac{1}{e^{\hbar\omega/k_BT} - 1}$$

Planck's blackbody spectrum:

$$u(\omega, T) = \frac{\hbar\omega^3}{\pi^2c^3}\frac{1}{e^{\hbar\omega/k_BT} - 1}$$

Energy density per unit frequency

Composite Particles

Statistics determined by total spin:

$^4$He atom: 2 protons + 2 neutrons + 2 electrons = 6 fermions → integer spin → boson
$^3$He atom: 2 protons + 1 neutron + 2 electrons = 5 fermions → half-integer spin → fermion
Cooper pairs: Two electrons (fermions) pair → spin-0 → boson (superconductivity)
Hydrogen molecule H₂: Even number of fermions → boson

Matter Structure

Fermions build structure:

Pauli exclusion → electrons in shells → atoms
No two electrons in same state → periodic table
Degeneracy pressure → stability of matter

Bosons mediate forces:

Photons → electromagnetic force
Gluons → strong force
W/Z bosons → weak force
Gravitons (hypothetical) → gravity

Key Differences Summary

Property	Fermions	Bosons
Spin	Half-integer	Integer
Wave function	Antisymmetric	Symmetric
Pauli exclusion	Yes	No
Max per state	1	Unlimited
Behavior	Spread out	Bunch together
Collective	Fermi gas	BEC, lasers

← Symmetrization Postulate Pauli Exclusion →

Quantum Mechanics Course