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The total wavefunction for identical fermions (electrons) must be antisymmetric under particle exchange: ψ(1,2) = −ψ(2,1). This is embodied in the Slater determinant. The Pauli exclusion principle (no two electrons in the same quantum state) follows as a consequence: if two electrons occupied the same state, the wavefunction would vanish.
From your study of jj-coupling and the Pauli exclusion principle, you know that no two electrons in an atom can share the same set of quantum numbers (n, l, m_l, m_s). Pauli introduced this rule in 1925 as an empirical observation to explain atomic spectra and the structure of the periodic table. But the exclusion principle is not a separate postulate bolted onto quantum mechanics — it follows from a deeper requirement about how the wavefunction of identical particles must behave when you swap their labels.
When two electrons are present, the quantum state is described by a wavefunction ψ(1, 2) — where labels 1 and 2 stand for the full set of coordinates (position and spin) of each electron. Because electrons are fundamentally indistinguishable — there is no physical label or dye that identifies "electron #1" versus "electron #2" — swapping the labels must leave all measurable quantities unchanged: |ψ(1,2)|² = |ψ(2,1)|². This means ψ(2,1) = ±ψ(1,2). Quantum field theory (and experiment) tells us which sign applies: electrons are fermions (spin-1/2 particles), so their wavefunctions must be antisymmetric under exchange: ψ(1,2) = −ψ(2,1). Bosons (integer spin, like photons) take the symmetric sign. This distinction — the spin-statistics theorem — is one of the deepest results in physics.
The Slater determinant provides a concrete way to build antisymmetric many-electron wavefunctions from single-particle states. For two electrons occupying single-particle states φ_a and φ_b, the properly antisymmetrized wavefunction is ψ(1,2) = (1/√2)[φ_a(1)φ_b(2) − φ_a(2)φ_b(1)]. Notice: if both electrons are in the same state (a = b), this becomes (1/√2)[φ_a(1)φ_a(2) − φ_a(2)φ_a(1)] = 0. The wavefunction does not just become small or improbable — it vanishes identically. A state with two electrons in the same single-particle state is not forbidden by fiat; it literally does not exist as a nonzero quantum state. The Pauli exclusion principle is an algebraic consequence of antisymmetry, not an independent axiom.
The physical consequences are vast. In atoms, antisymmetry forces electrons into distinct orbitals, building the periodic table shell by shell through the Aufbau principle — the chemistry of the entire universe rests on this. In metals, electrons fill all states up to the Fermi energy (forming a Fermi sea), producing electrical and thermal properties that classical physics cannot explain. In white dwarf stars and neutron stars, electrons (or neutrons) that cannot pile into the same state create degeneracy pressure — a quantum mechanical force that holds the star against gravitational collapse with no dependence on temperature whatsoever. The antisymmetry of fermionic wavefunctions, a symmetry requirement on a mathematical object, turns out to underlie the stability of all ordinary matter.