Bloch's theoremtheorem
In 1928, Felix Bloch, working on his dissertation under Heisenberg, pondered: how to describe an electron in a crystal where the atoms are arranged periodically? The answer turned out to be surprisingly elegant: the wave function must be 'Bloch-like' — the product of a plane wave and a function repeating with the lattice spacing. This naturally led to the idea of energy bands: some energy intervals are allowed, others forbidden. Thus band theory was born, explaining the difference between metals, dielectrics, and semiconductors.
How it works
Calculations of the band structure of all crystalline materials are based on Bloch's theorem. It explains electrical conductivity, optical properties, and allows engineers to design semiconductor devices: transistors, microchips, solar cells, and LEDs.
💡 Bloch left his mark in various fields: there are 'Bloch walls' in magnetism and the 'Bloch sphere' for qubits, but his theorem on waves in a crystal is a separate masterpiece.
\psi_{\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}} u_{\mathbf{k}}(\mathbf{r}), \quad u_{\mathbf{k}}(\mathbf{r}+\mathbf{R}) = u_{\mathbf{k}}(\mathbf{r})
ψₖ(r) is the electron wave function with quasi-wave vector k, uₖ(r) is a periodic function with the lattice period (same for each node), R is the lattice translation vector (shift by an integer number of periods), e ≈ 2.718 is the base of the natural logarithm, i is the imaginary unit, k is the quasi-wave vector (momentum divided by ħ), r is the spatial coordinate