The concept of the classical radius of an electron refers to an approximation based on classical electrodynamics. According to classical electrodynamics, if we treat the electron as a tiny charged sphere with its charge uniformly distributed, we can calculate its classical radius based on the balance between the electrostatic repulsion of the charged particles and the attractive forces holding the sphere together.
However, it's important to note that this classical model is an oversimplified representation and does not fully capture the nature of electrons according to modern physics. In reality, electrons are elementary particles, considered to be point-like particles with no size or spatial extent in the standard model of particle physics.
The idea that electrons are point-like particles arises from experimental evidence and the success of the theoretical framework that describes them. Numerous experiments have been conducted to probe the size and structure of electrons, and all measurements and observations to date are consistent with electrons having no discernible size or substructure.
In quantum mechanics, electrons are described by wavefunctions that are mathematical representations of their probabilistic behavior. These wavefunctions describe the likelihood of finding an electron at different locations, but they do not imply that the electron occupies a specific point in space.
It's worth noting that the classical radius of an electron is orders of magnitude larger than the theoretical predictions based on the standard model. This discrepancy further supports the notion that the classical model is not applicable at the microscopic scale of elementary particles like electrons.
In summary, while the classical model of the electron assigns a finite size to it, the modern understanding based on experimental evidence and quantum mechanics suggests that electrons are fundamentally point-like particles with no spatial extent.