The restriction to bosons and fermions in quantum mechanics arises from the fundamental principles of quantum statistics and the behavior of particles with half-integer and integer spin values. While it may initially seem arbitrary, there are underlying reasons for this restriction.
In quantum mechanics, particles are described by wave functions, which are mathematical representations that contain information about the particle's properties. The behavior of particles is governed by the wave function's symmetry under exchange of particles.
Fermions, such as electrons, protons, and neutrons, obey the Pauli exclusion principle, which states that no two identical fermions can occupy the same quantum state simultaneously. This leads to the observed phenomenon of electron shells and the stability of matter. Fermions have half-integer spin values (e.g., ±1/2, ±3/2, etc.) and are characterized by antisymmetric wave functions, meaning the sign of the wave function changes when particles are exchanged.
Bosons, on the other hand, have integer spin values (e.g., 0, ±1, ±2, etc.) and obey Bose-Einstein statistics. Unlike fermions, bosons can occupy the same quantum state simultaneously, leading to phenomena like Bose-Einstein condensation. Bosons are described by symmetric wave functions, meaning the sign of the wave function remains the same when particles are exchanged.
Now, coming to the question of why other spin values, such as ±1/5, do not exist, it's important to note that the spin of a particle is quantized, meaning it can only take certain specific values according to the rules of quantum mechanics. The spin of a particle is determined by its intrinsic angular momentum, which is a fundamental property. The allowed values for spin are determined by the symmetries of spacetime and the underlying mathematical structure of quantum mechanics.
The spin of a particle is always a multiple of ħ/2, where ħ (h-bar) is the reduced Planck constant. This fundamental property restricts the possible values of spin to half-integer or integer multiples of ħ/2. Consequently, the spin values of ±1/5, being fractional, do not conform to the allowed values of spin in quantum mechanics.
In summary, the restriction to bosons and fermions in quantum mechanics is a consequence of the underlying symmetries and principles of quantum statistics. The quantization of spin values and the resulting requirement for symmetric or antisymmetric wave functions lead to the observed behavior of particles in the quantum world. Other spin values, such as ±1/5, do not exist due to the fundamental nature of spin and the mathematical structure of quantum mechanics.