Blochfunktionen

Bloch functions, also known as Bloch waves or Bloch states, are mathematical descriptions of the wave functions of particles, typically electrons, in a periodic potential. They are fundamental concepts in solid-state physics and condensed matter physics, particularly in the study of crystalline materials. The name "Bloch functions" is derived from the Swiss physicist Felix Bloch, who introduced them in 1928.

In a crystalline solid, the potential experienced by an electron is periodic due to the regular arrangement of atoms. Bloch's theorem states that the wave function of an electron in such a periodic potential can be expressed as a plane wave modulated by a periodic function. Mathematically, a Bloch function is written as ψₖ(r) = e^(ik·r)uₖ(r), where ψₖ(r) is the wave function, k is the wave vector, r is the position vector, and uₖ(r) is a periodic function with the same periodicity as the crystal lattice.

The exponential term e^(ik·r) represents a plane wave, and uₖ(r) is a periodic function that reflects the crystal's symmetry. The wave vector k is a crucial parameter that determines the energy of the electron and is often referred to as the crystal momentum. The set of all possible wave vectors forms the Brillouin zone, which is a fundamental concept in the study of electronic band structures.

Bloch functions are essential for understanding various physical properties of solids, such as electrical conductivity, thermal conductivity, and optical properties. They provide a basis for the band theory of solids, which explains why some materials are conductors, insulators, or semiconductors. The concept of Bloch functions has wide-ranging applications in materials science, electronics, and nanotechnology.