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Fermi Statistics

The statistical mechanics describing the behavior of fermions in a system with a bounded potential, crucial for understanding nuclear properties.
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The statement of the theorem

Let n^i\hat{n}_{i} be the occupation number operator for a single-particle state ii. The Pauli Exclusion Principle dictates that the expectation value of the total number of fermions NN must satisfy the constraint: \n⟨N^⟩=∑i⟨n^i⟩≤g\langle \hat{N} \rangle = \sum_{i} \langle \hat{n}_{i} \rangle \le g \nwhere gg is the degeneracy of the state. The average occupation number ⟨n^i⟩\langle \hat{n}_{i} \rangle is given by the Fermi-Dirac distribution function: \n⟨n^i⟩=1e(Ei−μkBT)+1\langle \hat{n}_{i} \rangle = \frac{1}{e^{\left( \frac{E_{i} - \mu}{k_{B}T} \right)} + 1} \nwith μ\mu being the chemical potential and TT the temperature.