quantum mechanics - Spin Angular Momentum question

What does Spin Angular Momentum tell us? Why are we interested in knowing it? What does it mean physically when particles have spin as integers and half integers?

Answer

From one non-relativistic QM point of view, the Spin Angular Momentum appears as a source of magnetic moment intrinsic to matter, not related to its state of orbital motion, that ends up being one sort of angular momentum because of the algebraic properties it satisfies.

It describes one form of interaction of the particle with magnetic fields and this has quite important consequences. For example: it can account for ferromagnetism at the microscopic level.

The issue is that in QM spin isn't something that appears naturaly from the theory. It is included to satisfy observations such as the Stern-Gerlach experiment. One notices experimentaly that to account for such intrinsic magnetic moment one would need one angular momentum operator $\mathbf{S}$ with $S_x,S_y,S_z$ obeying the usual commutation relations, such that the only possible value of $S^2$ is $s = \frac{1}{2}$.

This yields a state space $\mathcal{E}_S$ for spin which is two dimensional generated by $|\pm\rangle$ with $S_z|+\rangle = \frac{\hbar}{2}|+\rangle$ and $S_z|-\rangle = -\frac{\hbar}{2}$. One then describes the particle with the tensor product $\mathcal{E}=\mathcal{E}_S\otimes \mathcal{E}_O$ with the state space describe the orbital degrees of freedom, namely, spanned by $|\mathbf{r}\rangle$ the position basis.

One then assumes also based on experiment that the magnetic moment is given by $\mathbf{\mu}=\gamma \mathbf{S}$ and builds the interaction Hamiltonian from it. So as I said, it is included in the theory by hand.

Now in QFT spin becomes a little bit more interesting. Particles are excited states of quantum fields. We then talk about the "spin of the field" meaning the spin of the particles associated to the field.

Integer spin fields are the so-called bosonic fields and its excitations are the so-called bosons. These are tensor fields, e.g., scalar, vector field, and so forth.

The half-integer spin fields are the so-called fermionic fields and its excitations are the so-called fermions. These are spinor fields.

From a mathematical point of view, all of these arise from the study of representations of the groups $SO(3)$ and $SO(1,3)$, respectively the rotation group and the Lorentz group. It is not inserted by hand, but it is a result of the Lorentz invariance of the theory itself, that there are particles which have this spin property. Furthermore, on the non-relativistic limit one recovers the QM approach.

Interestingly, it can be shown that the bosons obey the Bose-Einstein statistic when you treat systems with variable number of such particles, while fermions obey the Fermi-Dirac statistics.

The Fermi-Dirac statistic, for example, forbids in a system with more than one particle, that two particles be in the same state. Interestingly this is very important to explain the structure of the periodic table and hence of a lot of developments in chemistry.