symmetric wave function

In quantum mechanics, a wave function is a mathematical expression that describes the wave-like behavior of a particle. The square of the wave function, denoted as (\Psi^2), represents the probability of finding the particle at a specific point in space and time.12

For systems with identical particles, such as atoms with multiple electrons, the wave function must adhere to specific symmetry conditions. If the coordinates of two identical particles are interchanged, the wave function either remains unchanged (symmetric) or changes sign (antisymmetric). Particles with half-integral spin, like electrons, are called fermions and have antisymmetric wave functions. This leads to the exclusion principle, stating that no two fermions can occupy the same quantum state. Particles with zero or integral spin are called bosons and have symmetric wave functions.3