# Spin and the violation of the speed of light

## Main Question or Discussion Point

this is from Omnes' book Understanding QM

There was an older preconception according to which the electron was an electrically charged sphere with a radius of the order of 10^-15 m (for which the electrostatic energy is $m_ec^2$). If spin means a rotation of that sphere, the velocity at the sphere surface would have to be greater than the velocity of light. So accepting the idea of spin meant giving up the only model of the electron and, perhaps worse, the idea that angular momentum accompanies a rotational motion.
Has this problem been resolved or is it still outstanding?

## Answers and Replies

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CAF123
Gold Member
The spin of an electron is not meant in the literal sense (ie like some sort of spinning top or rotating sphere). Instead, it is a vector quantity and we define a spin quantum number.

I believe a sort of rough calculation goes as follows:
$$\tau = \frac{dL}{dt} = Iα$$
If we define $L = m_l\hbar = \frac{\hbar}{2}$ and $I = \frac{2}{5}mr^2,$ an electron as a solid sphere with $r ≈ fm$

Then, $$L = Iω = \frac{Iv}{r} => v = \frac{Lr}{I} = \frac{5Lr}{2mr^2}.$$
Simplifying and inputting known data gives v ≈ 1011 ms-1, which is 1000 times the speed of light.

So in other words the problem of the violation of the speed of light has not been resolved, right?

CAF123
Gold Member
The above calculation assumes that the electron is behaving like a rotating sphere and so it is meant to demonstrate why we should not think of spin as a classical concept here.

well if the electron is not spinning then what is it doing?

CAF123
Gold Member
Just to be a little clear on the question:
What do you mean by 'what do they do?'

Bill_K
well if the electron is not spinning then what is it doing?
The electron is observed to be pointlike, which means its size, if it has one, is smaller than we have so far been able to detect. (10-16 cm). As a particle, it has a number of properties: mass, charge and angular momentum. Angular momentum comes in two varieties: orbital angular momentum (r x p) and spin angular momentum. There is no motion associated with spin angular momentum, and so the electron does not rotate. The angular momentum it carries (ħ/2) is simply an intrinsic property.

mfb
Mentor
So in other words the problem of the violation of the speed of light has not been resolved, right?
The model (small, rotating sphere) which produced this prediction was wrong. Therefore, it is not a problem any more.

Spin is just another particle property, like its mass and charges (not just the electromagnetic charge).