Question

Which statistical distribution describes the behavior of identical, indistinguishable particles with half-integral spin?

• A. Maxwell–Boltzmann distribution
• B. Bose–Einstein distribution
• C. Fermi–Dirac distribution
• D. Gaussian distribution

Hint:

{Fermions (half-integer spin)} → Fermi–Dirac statistics {Bosons (integer spin)} → Bose–Einstein statistics

Answer 1

The Correct Option is C

Concept: Particles with half-integral spin (\(\frac{1}{2}, \frac{3}{2}, \dots\)) are known as fermions. Examples include electrons, protons, and neutrons. These particles obey the Pauli Exclusion Principle, which states that no two identical fermions can occupy the same quantum state simultaneously.
Step 1: Identify the appropriate statistics.} The statistical distribution that describes the behavior of fermions is the Fermi–Dirac distribution.
Step 2: Write the distribution function.} \[ f(E) = \frac{1}{e^{(E-\mu)/kT} + 1} \] where

• \(E\) = energy of the state
• \(\mu\) = chemical potential
• \(k\) = Boltzmann constant
• \(T\) = absolute temperature

Step 3: Compare with other distributions.}

• Maxwell–Boltzmann distribution applies to classical particles.
• Bose–Einstein distribution describes bosons (integral spin).

Thus, half-integer spin particles follow the Fermi–Dirac distribution.