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Pauli Exclusion Principle

No two identical fermions can occupy the same quantum state simultaneously, fundamentally shaping nuclear structure and dictating shell models.
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The statement of the theorem

Let ψi\psi_i be the wave function describing the ii-th fermion (e.g., proton or neutron) in a quantum state defined by quantum numbers (n,l,j,mj)(n, l, j, m_j). The Pauli Exclusion Principle dictates that for a system of NN identical fermions, the total wave function Ψ\Psi must be antisymmetric under the exchange of any two particles. Mathematically, this implies that the set of occupied single-particle states {ψ1,ψ2,…,ψN}\left\{ \psi_1, \psi_2, \dots, \psi_N \right\} must be distinct, such that for any two particles ii and jj (i≠ji \neq j), the state ψi\psi_i cannot be identical to ψj\psi_j. Formally, the occupation number operator n^k\hat{n}_k for any given state kk must satisfy n^k∈{0,1}\hat{n}_k \in \{0, 1\}. The Hamiltonian for the system is then constructed using second quantization: H^=∑kEkn^k\hat{H} = \sum_k E_k \hat{n}_k.