The short answer is: Electron spin cannot be understood in its classical meaning.
The spin of an electron is the degree of freedom. It is similar to position and momentum. If a particle’s momentum is zero, its force is zero. In the same way, as long as there isn’t torque on the system the spin is also conserved.
But wait! But wait!
Technically: The spin of a particle is conserved as long as the Hamiltonian does not contain terms that are related to the spin degree.
It might sound like Hah! The electron spin behaves classically, however! It doesn’t. It is not classically imagined by physicists for these reasons:
An ordinary object spinning on its axis will have an angular momentum. This is determined by the distribution of mass around the axis and the speed at which the object spins.
Fixed angular momentum means that the mass distributed further from the axis will have a lower angular velocity. However, mass distributed closer to the center of the axis will have a higher angular speed. Imagine a spinning ice skater turning at a slower speed with his arms extended, and at a faster speed with his arms pulled up overhead (on the Axis).
We can extend the analogy to electrons. Strange things can happen!
An electron of finite size, which is too small to see, can cause problems in classical descriptions. This includes charge self-repulsion with infinite energy and surfaces that spin faster than the speed light.
My personal bias against interpreting an electron in a classical sense is that if we try to model it as a charged sphere spinning along its axis and this spinning is what causes its magnetic moment, then one could quickly calculate that the surface would move at speeds far greater than the speed light!
You can look at the figure to see more details (or skip to the formula at end).
The following results can be obtained by using the calculations above:
Which is the same as
Entering the value of electron magnetic moment = (-9.285 * 10(-24),)
Charge on an electron = 1.6*10(-19).
Even if R is as large as the Classical electronic radius,
= 10-15, the value of is on the order 1028! This is orders of magnitude faster that the speed of light (which is only order 108)!
We have also not seen any scattering experiments that reveal a composite nature of an electron. However, we have established the lower bound for the electron size through several experiments. This, if we plug in the above equation, will just make the speed increase! Hans Dehmelt has more information.
The classical description of an electron spinning charge as a spinning one seems to be very easy to disintegrate.
But, the electron cannot have angular momentum if it is a point particle. This means that it has no finite size. The entire object is spinning around its rotational axis so the electron can’t spin about its center of mass. How can we get out of this dilemma?
It is impossible to “see” an electron in order to tell if it is spinning. It is impossible to measure the “spinning”, so it does not make sense to talk about it in science. You can measure the electron’s angular motion; science makes it logical to talk about angular momentum. Don’t view the electron as a spinning object, which we cannot observe or know; instead, think of it simply as having “intrinsic” angular momentum.
Here are some technical details:
Classically, thinking of spin as “spinning”, introduces problems conceptually because it doesn’t generalize well to massless objects and doesn’t handle half integer spin well. It also makes it difficult to interpret electrons correctly, aside from feeling like you know the source of the angular momentum.
However, thinking of spin in terms of intrinsic angular momentum is a way to avoid all of the issues mentioned above and, as you will see with more study, it fits well into the rest of physics.
Let me end with a final word
Concerning spin-1/2, it is a result of group theory. It can’t really be explained in a post. Study some group theory for scientists and ask more questions!
This is it folks!
I will add an additional calculation to show how you can artificially reproduce the classical gyromagnetic relationship of an electron.
Let and be the mass of the electron and the charge densities. This is the factor that gives:
Consider exponential densities as a result of some screening at a fundamental level.
From which we find; if you take, we get. Although this is artificial, there may be an electrostatic model that can accommodate it.
The classical spin interpretation has many issues that can be easily solved in the quantum mechanical interpretation. We therefore choose the latter.