Although angular momentum was initially a classical concept in physics, quantum physics allowed physicists to find a new definition that was comparable to the original one. This was back when quantum features were not important.
For states in which electrons orbit nuclei, the first quantum definition for angular momentum was discovered. This was the first orbital angular momentum. It was created using the rules of quantum Physics by the operator pxr. p is the momentum and r the position operators. This is identical to the classical Physics definition, except that pxr and r are operators instead of numbers. This similarity means that in cases where the quantum effects are very small (e.g. They would “correspond to” the classical values in situations where quantum effects are small (e.g., large orbits). This “correspondence principle” was a key element in the development and application of quantum physics. It basically said that the limit of new quantum physics could be interpreted to classical physics.
Then, there was a problem. This orbital angular momentum wasn’t conserved! For example, it could shift by 1 unit when a photon is emitted. This could be explained by considering that photons have intrinsic angular momentum. A circularly polarized light wave of light has angular momentum. This is consistent with the correspondence principle.
Even more mysterious was the fact electrons could have intrinsic angular momentum even when at rest. This fascinating history begins with the discovery of it by George Uhlenbeck, Samuel Goudsmit and the incorporation of that spin by Dirac into quantum physics equations. Many experimentalists are still puzzled as to why this discovery wasn’t awarded a Nobel Prize. It is due to a huge oversight by the Nobel committee.
Although it is difficult to explain the spin of an electron, beginning students are told to imagine a small and massive object with some extent rotating around its axis. This is not the true quantum description for an electron. It is a point-like particle in quantum physics. It could one day be demonstrated to have a structure. String theorists claim that they can gain insight by giving it some extent in 10 spatial dimension; however, this theory is not yet proven to be true.
An interesting fact about electron spin is that you can precess it by placing the electron in a magneticfield. Precess the electron 360 degrees to make it orient in the same direction. Quantum theory says that the wave function for this particle is different from the one representing the original unprecessed electron. It is multiplied with a minus sign. This fact is not analogous in any classical way. Experimentally, the minus sign was observed by interfering with an electron in a two-slit diffraction experiment. Incredible! It’s true.
Get specific answers to your questions
(a) How does a wave spin around its axis.
It doesn’t. It has angular momentum. QM doesn’t consider a wave spinning.
(b) What caused the electron spin and not lose spin momentum/velocity?
It was born this way. It isn’t spinning if it doesn’t spin. Super-symmetry theories state that there is a non-spinning electronic. It is called a selectron. (Get it? s-electron.) However, experiments to discover such particles have been unsuccessful for over 40 years. Burt Richter pointed out that Moses took 40 years to find the promised Land. He succeeded. He believes it is time to let go of supersymmetry theorists.
(c) What caused each electron to spin at exactly the same speed as s =+/-1/2?
Quantum theory and the symmetry it exhibits regarding rotations states that all fundamental objects cannot have spins greater than 1/2. This includes zero and one.
(d) Do free electrons have zero orbital momentum, and the same s as when they were free before?
No. No. The origin of the axes will determine the orbital angle momentum of a free electron. An electron that is ejected from anatom often contains orbital momentum. You can picture this as a pitcher throwing a baseball. He releases the electron at about one meter from his center of gravity. The same is true for photons emitted by an atom. They too often carry orbital angular momentum.
An electron can retain its original spin or undergo a spin flip after being released. In this case, its spin is changed by 1 unit.