The mechanistic representation (not really an interpretation ;-)) does indeed not make any sense. I am curious though how you come to the linear speed of a point on the surface of the sphere at 1 million times c, that’s 3 x 1014 m/s. I back calculated, that means rotation speed would be 1030 rpm and a centripetal acceleration of around 1047 m/s2! Lol!
My personal representation of spin is… no representation… It is just a state, like color for gluons, or even happy or sad for human beings. No geometry is involved… But it does help when dealing with symmetries…
And for a particle, it is the same, I do not see it as a particle, I do not see it as a wave… I do not even see them as a variation in a field in a physical sense. I see these more like the variation of a mathematical function in space-time (I am from the school that believes that reality is an emergent property of a mathematical construct)… I know it is weird, it is a little like thinking that ideas are real, and reality as we know it results from them… But it is the only way my (Brownian ?) brain can make sense of it all…
Yet, these physical representations are really practical to understand and predict things. But they do have limits. The caution is to know at what point one needs to back off, and allow a more abstract view. And that’s where I regret having had bad a terrible maths teacher in high school and consequently not taking an advanced maths course at Uni ;-).
Take an angular speed ω around the z axis. A point located at a distance r from the center (in spherical coordinate) lies at a distance r sin θ from the z axis. It will therefore rotate with a velocity ω r sin θ.
Let us now calculate the angular momentum. We need to integrate over a sphere of radius R and constant density ρ the quantity ω ρ r2 sin2 θ dV. This gives
L = 2/5 m v R
where m is the electron mass and v=ω R.
Now, for having this being associated with a spin of 1/2, we must also have
L = 1/2 hbar.
The rest is easy and you get v = 5/4 hbar / (m R) = 1.45 1014 m/s.
This is what is important. We need to keep in mind the limits of the representations. I hope we will meet at Virgo in September :)
Thank you @leMouth, it made me revise my moments of Inertia :-).
I would love to come to Virgo in September! Yet, It will corresponds to the second week of the scholar year, a time where my agenda is a just a big mess... not the best period to plan things ahead. Still, I am reflecting if I could manage it as it could be quite something!
Fingers are crossed! :D