Yet it does not really rotate on itself, not in the classical sense of the term. For a spinning top or a dancer, from a certain frame of reference, we can observe a body rotating around an axis ideally going through its centre of mass. But that doesn’t work for the electron.
In general, a rotation around its own axis implies the presence of some form of energy (for example, mass) extended in space, which runs a circular motion with respect to the axis itself. The electron, however, does not have an extension: although it is provided with a certain distribution of probability of being in one point of space instead of another, it is considered a point object (for all practical purposes, you would never be able to locate where an elementary particle begins, or ends). And you can’t, of course, measure the rotation of a material point.
This is also the case with other quantum properties. If these themes are new to you, I understand that maybe at the beginning it might seem strange to you to know that scientists describe quantum particles, and not only that, through properties that you can’t create some sort of visual image of - like, Conversely, you are generally used to making for everyday life objects.
In short, the analogies between the microscopic and the macroscopic world are particularly limited. In other words, you can’t imagine an electron spinning just because in the past you’ve experienced a skater twirling on ice. You can't really represent the spin graphically, because in the elementary particle the concept of view (with any tool) loses its meaning.
However, even assuming, absurdly, that the electron revolves around itself, it would appear that the points on its surface (assuming this exists, which we have already ruled out) would move faster than light in a vacuum. This result, however, would contradict the already well-proven theory of Special Relativity, presented by Albert Einstein in 1905.
Why, then, did they call it "spin"?
Its introduction dates from the first half of the last century by the well-known physicist Wolfgang Pauli in 1924. He, in an attempt to explain the structure of atomic spectra and the periodic table, had recently postulated that electrons were also to be characterized by a physical magnitude capable of assuming only two distinct values, but that was not classically describable.
It was only known that this new quantity had algebraic properties, in some ways, similar to those of a well-known classical magnitude: the so-called angular momentum (it describes the state of rotation of a body in the classical regime; typically for macroscopic objects). It was thus decided to name it "spin" for a mere analogy - only at the level of mathematical formalism - with angular momentum.
Sources: Scientific American, Cambridge.
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