A point charge

The electric field E at a distance r from the electron is equal to $e\mathrm{/}4\pi {ϵ}_0{r}^2,e/4\backslash pi\backslash epsilon\_0r^2,$ and becomes infinite when r is zero. This is troublesome. The difficulty was already present in Laplace's equation, which determines the electric potential $\mathrm{\Delta }\text{V}=0\backslash Delta\; \backslash text\{V\}\; =\; 0$, where $\mathrm{\Delta }\text{V}={\displaystyle \frac{{\mathrm{\partial }}^2\text{V}}{\mathrm{\partial }{x}^2}}+{\displaystyle \frac{{\mathrm{\partial }}^2\text{V}}{\mathrm{\partial }{y}^2}}+{\displaystyle \frac{{\mathrm{\partial }}^2\text{V}}{\mathrm{\partial }{z}^2}}\backslash Delta\; \backslash text\{V\}\; =\; \backslash dfrac\{\backslash partial\; ^2\; \backslash text\{V\}\}\; \{\backslash partial\; x^2\}\; +\; \backslash dfrac\{\backslash partial\; ^2\; \backslash text\{V\}\}\; \{\backslash partial\; y^2\}\; +\; \backslash dfrac\{\backslash partial\; ^2\; \backslash text\{V\}\}\; \{\backslash partial\; z^2\}$

denotes the Laplacian. The problem was solved by distribution theory, with the point charge represented by Dirac's "function" δ : $\mathrm{\Delta }\text{V}=e\delta (r)\mathrm{/}4\pi {\epsilon }_0\mathrm{.}\backslash Delta\; \backslash text\{V\}\; =\; e\; \backslash delta\; (r)\; /\; 4\backslash pi\; \backslash varepsilon\; \_0.$

Similar difficulties arose when spin was introduced. It was intended to explain the fine structure of hydrogen spectral lines and the result of the Stern–Gerlach experiment, in which a beam of electrons is split into two parts by a non-uniform magnetic field. In 1925, George Eugene Uhlenbeck and Samuel Abraham Goudsmit proposed assigning the electron a magnetic moment of 1/2, which explained the observations. When Uhlenbeck presented this model of the electron to his mentor, the physicist Paul Ehrenfest, he spotted an impossibility. Before quantum mechanics, a radius was assigned to the electron by equating its electrostatic energy with its rest energy (via E = mc ² ): r = e ²/(4πε₀ mc ² ). The electron had to rotate about its own axis for its charge to generate a magnetic field. Ehrenfest calculated that parts of the electron would be moving faster than light! Thus, no one knew how a rotating point charge could generate a magnetic field, and assigning it a radius did not solve the problem. Fortunately, Ehrenfest argued that Uhlenbeck was young enough to make mistakes and encouraged him to publish his hypothesis; quantum mechanics vindicated him. Thus Uhlenbeck invented electron spin.