In the late 1920s, physics faced an unresolved paradox about the behavior of electrons inside a metal. On one hand, Sommerfeld's free-electron model — treating conduction electrons as a gas moving unobstructed within the metal — reasonably explained some thermal and electrical properties, but completely ignored that electrons actually move within a periodic crystal lattice of ions. On the other hand, treating each electron as bound to its parent atom explained the behavior of inner electrons, but did not work for conduction electrons that clearly move throughout the entire crystal. Felix Bloch, a doctoral student of Werner Heisenberg in Leipzig, resolved this tension in the summer of 1928 with an idea combining both worlds: he mathematically showed that, if the electric potential an electron experiences is periodic — as it necessarily is in any ordered crystal lattice — then the solution of the Schrödinger equation for that electron must take the form of a plane wave modulated by a function sharing the same periodicity as the lattice. These solutions, today called Bloch functions, are neither Sommerfeld's fully free electron nor the fully bound electron of an isolated atom, but an exact mathematical synthesis of both limits. The immediate consequence, which Bloch himself developed in the same thesis, is the existence of allowed and forbidden energy bands (gaps) for the electrons of a crystal — the conceptual framework that, developed shortly after by Alan Herries Wilson, would for the first time rigorously explain why some materials are metals, others are insulators, and others, semiconductors, occupy a crucial intermediate ground. Bloch's theorem is the obligatory starting point of all subsequent band physics — including the calculation techniques of Slater, Herring, and their successors — and the ultimate theoretical basis of all modern solid-state electronics, from the transistor to the microprocessor.