Part 2 of 4: Spin

Fundamental particles have a number of intrinsic properties, which are fixed, and the same for all members of a species. These include examples such as the rest mass of a particle, and its electric charge. Another intrinsic property is one called ’spin.’ The spin is a vector (of sorts), and its direction for particles of non-zero spin is an intrinsic degree of freedom.
Recall our discussion of the angular momentum in quantum mechanics, particularly the operators and their commutation relations, and the eigenvalue ladder. Spin is often described as being like the particles rotating around their axis; this fits only in that the previously mentioned mathematical description of quantized angular momentum also applies to spin, and that spin must be counted in the total angular momentum, with a few powerful differences; in analogy to the angular momentum numbers l and m (, ), we have s and m_s (, ). In one difference, the relationship between a charged particle’s spin and its magnetic moment involves a g-factor, and thus a gyromagnetic ratio, incompatible with classic physics theory (see here).

Another important difference is in the allowed values of s. As we found here, m ranges in integer steps from -l to l, so that for integer n; similarly, m_s ranges in integer steps from -s to s. When dealing with the eigenstates of angular momentum in position space, continuity in the azimuthal angular coordinate eliminated the half-integer values for l. Spin, however, is under no such restriction; s may be an integer or a half-integer.

I stated earlier that spin is an intrinsic property of a particle; in particular, the value of s is fixed for a given particle species. For example, electrons, protons, neutrons, and all flavors of quarks are all spin-½, which means s=½ for these particles. Photons are spin-1 (associated with polarization), while the delta baryons have a spin of 3/2. The hypothetical graviton, if it exists, must be spin-2 (as conservation of mass and momentum require that gravitational wave mechanicss by quadrupole wave mechanicss), and the hypothetical Higgs boson would be the only elementary particle with a spin of zero.