Well, before doing axiomatics, which is a very good thing after having some intuitive ideas about an subject, but doesn't help much to get the physics underlying the theory. That's particularly important in quantum theory, which is very abstract compared to all of classical physics (classical="not quantum" here, relativity is included in classical physics as long as it's not quantum
.
There's also a problem with the didactics, because many books start with a kind of historical heuristics, which however confuses students more than it helps, because the socalled "old quantum theory" (1900-1925) is outdated and gives wrong pictures on nature even on a qualitative level.
That's why I recommend as a first book
J.J. Sakurai, Modern Quantum Mechanics (2nd edition), Addison Wesley
It starts with a thorough analysis of the Stern-Gerlach experiment and spin observables. This is the most simple non-trivial example. So let me sketch it here.
It's an experimental finding by Stern and Gerlach in one of the most important experiments ever conducted in the history of science. It was done in 1923, i.e., before "modern quantum theory" was discovered by Heisenberg, Born, and Jordan (1925), Schrödinger (1926) and in its final formulation by Dirac (1926). Here, I follow the modern version a la Dirac.
The first observation is that particles like and electron have intrinsic angular momentum, called spin. It's very hard to imagine when trying to deal with it in a classical picture. Thus one shouldn't even try this! Phenomenologically the spin manifests itself as a magnetic moment of a particle like and electron or a neutron (the experiment was actually done with neutral silver atoms). This is familiar from classical electrodynamics: A current distribution looked from some distance looks in first approximation like a socalled dipole field. You can forget about the concrete current distribution and substitute it by a "point dipole", i.e., a point singularity in the current distribution at a certain place and the corresponding magnetic field it creates. Also such an "elementary magnetic dipole" is affected by an inhomogeneous magnetic field, i.e., it feels a force like a magnet (which is, looked at some distance to a good approximation such a dipole) in an inhomogeneous magnetic field.
Now, when Stern and Gerlach run a beam of silver atoms, prepared by heating up silver in and oven and let out a beam through a small opening, through an inhomogeneous magnetic field and then detecting them on photo plates they found a surprising result that could not be explained with classical physics: The beam of silver atoms did not show some continuous spread as to be expected from a classical set of dipoles which are randomly oriented due to the thermal motion of the silver atoms inside the oven, but they found a split in two clearly separated lines. There was no way to predict in which of the two branches of the beam an individual silver atom might occur, but it were two discrete spots on the screen!
The modern explanation is as follows: The magnetic moment of an electron is quantized, i.e., letting it run through a magnetic field with a large (quasi homogeneous) magnetic field in ##z## direction plus some inhomogeneous field in another direction there are two possible values for the spin and, proportional to it, its magnetic moment. As angular momenta the spin component is measured in units of the modified Planck constant, and for an electron the possible values of its spin-z component are ##s_z=\pm \hbar/2##. The direction of the large homogeneous part of the magnetic field determines the spin to point in z-direction (at least on average, we'll come back to this) and the inhomogeneous piece leads to the corresponding force on the electron. This explains, why there is not a continuous line on the screen as predicted by classical mechanics but only two spots.
Now, using another Stern-Gerlach experiment of the same type (i.e., one that measures the z-component of the spin) on one of the partial beams, no more splitting of the beam is found, but it always comes out the spin-z value of the chosen partial beam.
If however, the 2nd Stern-Gerlach experiment is set up to meausure a component of the spin in perpendicular direction, say in x direction, again you find two spots, and it's impossible to predict for an individual silver atom which value of ##\sigma_x =\pm \hbar/2## you'll measure. There's only a probability of ##1/2## to find ##+\hbar/2## and a probability of ##1/2## to find ##-\hbar/2##.
This is pretty analogous with the well-known phenomenon of polarization of the electromagnetic field, which is described by the classical electromagnetic waves predicted by Maxwell. After some work on the foundations, the quantum physicists came to the following set of rules (the axioms bhobba is talking about), which I formulate for spin here.
(1) The spin state of an electron is described by vectors in a two-dimensional complex Hilbert space. That's a two-dimensional vector space with complex numbers as scalars, which has a scalar product.
(2) The spin components are represented by self-adjoint operators ##\hat{s}_j##, ##j \in \{x,y,z \}##, which obey the commutation relations of the Lie algebra of the rotation group, i.e., they represent infinitesimal rotations in three-dimensional Euclidean space. From this one can derive the commutation relations
##[\hat{s}_j,\hat{s}_k] = \mathrm{i} \hbar \epsilon_{jkl} \hat{s}_l.##
(3) The possible outcome of a measurement of the j-component of the spin are the eigenvalues of the corresponding operator ##\hat{\sigma}_j##. One can find the eigenvalues and the structure of the Hilbert space by analyzing the commutation relations carefully. It turns out that you can specify the values of ##\vec{s}^2## and one spin component, usually one takes the z-component, simultaneously, because the corresponding operators commute:
##[\hat{\vec{s}}^2,\hat{s}_z]=0.##
Then it comes out that the eigenvalues of ##\hat{\vec{s}}^2## are ##\hbar^2 s(s+1)## with ##s \in \{0,1/2,1,3/2,\ldots \}##. For a given eigenvalue of ##s## (which determines the spin of the particle) the z-component of the spin can take the values ##s_z=\hbar \sigma_z## with ##\sigma_z \in \{-s,-s+1,\ldots,s-1,s \}##, i.e., for each ##s## there are ##(2s+1)## eigenvalues of ##\hat{\sigma}_z##. From Stern and Gerlach's experiment thus an electron must have spin ##s=1/2##, so that there are exactly two possible values for ##s_z \in \{-1/2,+1/2 \}##. The eigen vectors are called ##|s_z \rangle##. These vectors are assumed to be normalized, and they are orthogonal to each other, because according to a theorem from linear algebra (or Hilbert space theory, which here is the same because we deal with a finite-dimensional Hilbert space) the eigenvectors of self-adjoint operators with different eigenvalues have perpendicular eigenvectors.
(4) Filtering out the partial beam corresponding to the value ##s_z=+1/2##, i.e., ##\sigma_z=\hbar/2##, leads to silver atoms whose valence electron is in the state represented by ##|s_z=+1/2 \rangle##.
(5) The physical meaning of these state vectors is the following: When measuring any observable (e.g., the spin-x component) on a beam of particles prepared in the state represented by ##|s_z=+1/2 \rangle##, then the probability to measure a certain value of ##\sigma_x = \pm \hbar/2## is given by
$$P_{s_z=1/2}(s_x)=|\langle s_x|s_z=1/2 \rangle|^2.$$
As the calculation for this case shows, one finds indeed the observed values:
$$P_{z_z=1/2}(s_x=+1/2)=1/2, \quad P_{s_z=-1/2}(s_x=-1/2)=1/2.$$
This implies that for an electron with definite ##s_z## component the ##s_x## component's value is totally unknown and vize versa. That's because, there are no common eigenvectors of ##\hat{\sigma}_z## and ##\hat{\sigma}_x##, because these operators don't commute: ##[\hat{\sigma}_z,\hat{\sigma}_x]=\mathrm{i} \hbar \hat{\sigma}_y \neq 0##.
