Particles with whole-number spin are called bosons, while particles with half-integer spin are called fermions.
Bosons have integer values for their spin, such as 0, 1, 2, and so on. Examples of bosons include photons (spin 1), W and Z bosons (spin 1), and the Higgs boson (spin 0). Bosons follow Bose-Einstein statistics and can occupy the same quantum state simultaneously, leading to phenomena like Bose-Einstein condensation.
Fermions have half-integer values for their spin, such as 1/2, 3/2, 5/2, and so on. Examples of fermions include electrons (spin 1/2), quarks (spin 1/2), and neutrinos (spin 1/2). Fermions follow Fermi-Dirac statistics and obey the Pauli exclusion principle, which states that no two identical fermions can occupy the same quantum state simultaneously.
Regarding the symmetries of the wave function, bosons have symmetric wave functions under particle exchange. This means that if you swap the positions of two identical bosons, the wave function remains unchanged. In mathematical terms, the wave function is symmetric under the exchange of identical particles.
On the other hand, fermions have antisymmetric wave functions under particle exchange. If you swap the positions of two identical fermions, the wave function changes sign. In mathematical terms, the wave function is antisymmetric under the exchange of identical particles.
These symmetries of the wave function play a crucial role in determining the behavior of particles and the properties of their systems, such as the formation of atoms and the structure of matter.