antisymmetric wavefunction

An antisymmetric wavefunction is a multiparticle state that flips its sign when you exchange two of the particles. Swap the labels of particle one and particle two and the whole expression becomes its own negative — same magnitude, opposite sign. Since the squared magnitude is what we measure, the observable predictions are untouched, exactly as indistinguishability demands; only the unobservable phase changes. This is the exchange behaviour that defines fermions.

You build such a state by subtracting the two label arrangements instead of adding them. For two particles in states a and b, the antisymmetric combination is 'particle one in a, particle two in b' minus 'particle one in b, particle two in a'. The minus sign carries a striking consequence: if you try to put both particles into the very same single-particle state, the two terms become identical and cancel, leaving nothing. That is the Pauli exclusion principle emerging directly from the structure of the state.

The same minus sign also keeps identical fermions apart in space. Two electrons described by an antisymmetric state are less likely to be found close together than distinguishable particles would be — an effective avoidance sometimes called the exchange hole. This statistical standoffishness, demanding no real force between them, helps shape the energies of atoms and the magnetism of materials, and it is why fermionic matter is stiff rather than collapsible.