antisymmetrisaation

Antisymmetrisation is a process used in quantum mechanics to ensure that the wavefunction describing a system of identical fermions obeys the Pauli exclusion principle. The Pauli exclusion principle states that no two identical fermions can occupy the same quantum state simultaneously. In the context of a multi-particle wavefunction, this means that swapping the coordinates of any two identical fermions must result in a sign change of the wavefunction.

To achieve this property, the total wavefunction of a system of identical fermions is constructed as an antisymmetrized linear combination of the wavefunctions of individual particles. This is typically done using a Slater determinant. A Slater determinant is a specific type of determinant formed from single-particle wavefunctions, where each column represents a particle and each row represents a quantum state. The determinant naturally incorporates the antisymmetry requirement. If any two rows (representing two identical particles) are identical, the determinant becomes zero, reflecting the impossibility of two identical fermions being in the same state.

The mathematical operation of antisymmetrisation can be represented by an antisymmetrisation operator, which acts on a wavefunction and ensures its antisymmetry with respect to the exchange of any pair of identical particles. This is a fundamental concept in quantum chemistry and condensed matter physics, essential for describing the behavior of electrons in atoms and molecules, as well as other fermionic systems.