I disagree with the statement quantum spin would have nothing to do with classical spin. Of course, as usual, one must not use classical analogies to quantum theory in a careless way. The intuition on the behavior of quanta gained by learning modern quantum theory (the only consistent and very successful theory we have about the behavior of matter today) is necessarily not in a direct sense from extrapolating our everyday experience with macroscopic matter that behaves according to the laws of classical physics.
So to understand "spin" it's indeed pretty misleading to think about, say, an electron as a little classical ball that is rotating. Rather an electron is described by a quantum field.
To get an idea, how "spin" is described in quantum theory we must get an idea from classical physics that is adequate to be generalized to the quantum realm. There is no strictly logical path to this, because quantum theory is more comprehensive than classical mechanics, which is an approximate description of quantum mechanics for macroscopic quantities that can be understood as coarse grained averages over a lot of microscopic degrees of freedom. So there is always some heuristics involved when one likes to "quantize" a classical model.
It has turned out that the key tool to understand this "quantization procedure" are symmetry principles. In physics by a symmetry we mean some transformation of a set of observables that keep the dynamical equations for this observables unchanged. It's like symmetry in geometry: You can rotate a sphere around its center around any axis by any angle (that's the operation) without changing it in any way (that's making the operation to a symmetry).
Very important symmetries are those of the space-time description. Here we stick to the non-relativistic (Galilei-Newton) space-time description. It describes space as a three-dimensional Euclidean manifold, known from school geometry, and time as a directed line. Space is symmetric under translations, i.e., the geometry is everywhere in space the same (no distinguished point) and under rotations around any point (no direction is distinguished in any way). Further, the fundamental physical laws do not change with time, i.e., physics is also invariant under time translations. Finally, we can not determine any dependence of the physical laws on the absolute velocity against any fixed reference frame, i.e., there exists the class of inertial frames, where Newton's 2nd Law holds (a body moving freely will move in a straight line with constant velocity or stay at rest), and one cannot distinguish the motion with constant speed of one such inertial frame against another one. The whole set of symmetry transformations build a mathematical structure called a group, i.e., with two transformations also their composition is again a symmetry transformation and for each symmetry transformation we can find another symmetry transformation bringing us back to the original situation.
In classical mechanics the most adequate description of all these symmetry ideas is to work in phase space, i.e., for a set of N point particles you use their positions and momenta to describe their behavior, which makes together 3N position coordinates and 3N momentum coordinates. So phase space is a 6N dimensional space. The equations of motion can be derived from Hamilton's principle of least action, leading to the socalled Poisson brackets. The change of a quantity in phase space with time is governed by equations like
\mathrm{d}{\mathrm{d} t}F(\vec{q},\vec{p})=\{F(\vec{q},\vec{p}),H(\vec{q},\vec{p}) \}=\frac{\partial F}{\partial \vec{q}} \frac{\partial H}{\partial \vec{p}}-\frac{\partial F}{\partial \vec{p}} \frac{\partial H}{\partial \vec{q}},
where H is the Hamilton function, given the total energy of the system as function of the phase-space coordinates.
Then a symmetry can be described as an action on the phase-space variables, which leaves the fundamental Poisson brackets
\{q_j,q_k \}=\{p_j,p_k \}=0, \quad \{q_j,p_k \}=\delta_{jk}
invariant, the socalled canonical transformations.
Applied to the symmetry transformations of spacetime this leads to Noether's theorem: Any continuous symmetry is generated by some phase-space quantity, and that quantity is conserved. In this way one is lead to the definition of the conserved quantities by their role as generators of the corresponding symmetry transformations. For Galilei-Newton spacetime that's
temporal translations -> energy
spatial translations -> momentum
spatial rotations -> angular momentum
Galilei boosts -> center-of mass position
The Galilei boosts are a bit special, but we don't need to discuss them further, although that's a pretty interesting subject in itself.
Here we concentrate on angular momentum and rotations. In quantum theory we have a pretty similar structure although that's not so obvious on first glance. First of all the (pure) states are represented by rays in a Hilbert space, and the observables by self-adjoint operators on Hilbert space. One way to define the observables is to look for realizations of the symmetry group of space-time in terms of such self-adjoint operators as generators for the corresponding unitary transformations that represent the symmetry transformations, building a socalled unitary representation of this group. Now there's a entire mathematical theory behind such group representations by unitary operators on Hilbert space.
Now you can start simply with "quantization" of a classical particle moving freely by introducing operators for position and momentum and demand that instead of Poisson brackets the corresponding relations hold for the commutators (up to a factor \mathrm{i}):
[\hat{q}_j,\hat{q}_k]=[\hat{p}_j,\hat{p}_k]=0, \quad [\hat{q}_j,\hat{p_k}]=\mathrm{i} \hbar \delta_{jk}.
As it turns out this is consistent with the idea that the momentum operators are the infinitesimal generators for translations in space, and one can build the entire quantum theory for a particle on this basis. The angular-momentum operators, defined as generating rotations around the corresponding axes, are simply given as in classical physics by \vec{L}=\vec{x} \times \vec{p} (here as an operator relation).
Now this is, however, not the only possibility! Quantum theory admits somewhat more sophisticated realizations of the Galilei symmetry, in that in addition to the above given orbital angular momentum \vec{L} there can also be an additional spin-angular momentum. To understand its meaning one must refer to the fact that unitary operators of course act on the Hilbert-space vectors and not only as unitary transformations on the operators describing observables. Now consider a particle at rest. The corresponding vector in Hilbert space would be a momentum eigenstate with eigenvalues 0 for all three momentum components. Such a thing doesn't really exist but only as a socalled "generalized eigenstate", which is not normalizable to 1 but only to a Dirac \delta distribution, but this we neglect here just as physicists use to do it.
The point is that there exist unitary representations of the Galilei group where the generalized momentum eigenstates for vanishing momentum change under rotations. The generators for these rotations introduce new observables, unknown for classical point particles, that behave like a contribution to total angular momentum, which now is given as
\vec{J}=\vec{L}+\vec{S}.
In nonrelativistic quantum theory the spin operator commutes with momentum, and thus one can diagonalize also \vec{S}^2and S_z together with momentum. An &quot;elementary&quot; particle is described by definition by an irreducible representation of the Galilei group, and thus one must have a fixed eigenvalue ofr \vec{S}^2, which must take the value s(s+1) \hbar^2 with s \in \{0,1/2,1,3/2,\cdots \}. For s=0 the spin components are also always 0, and we are back to a particle described in the naive way by quantization the classical particle. For s \neq 0 a particle at rest can have also a determined s_z spin component with s_z=\sigma_z \hbar with \sigma_z \in \{-s,-s+1,\ldots,s-1,s \}.<br />
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Usually a particle with spin carries a magnetic moment that is described by the operator<br />
\vec{\mu}=\frac{q}{2m} g \vec{S}.<br />
Here q is the charge of the particle, m its mass, and g the socalled gyromagnetic factor (which is close to 2 for an electron but can take different values for composite particles like the proton). <br />
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In this way spin was observed for the first time in the famous Stern-Gerlach experiment, although at this time the above described modern notion of spin was not known, and it has been misinterpreted using old quantum theory (the Bohr-Sommerfeld model for atoms). Funnily enough there are two mistakes in this outdated model compensating each other such that it could be interpreted as proving the &quot;quantization of the direction&quot; predicted by the Bohr-Sommerfeld model of the magnetization of the electron.
