Hofstadter's butterfly
Videos
In condensed matter physics, Hofstadter's butterfly is a graph of the spectral properties of non-interacting two-dimensional electrons in a perpendicular magnetic field in a lattice. The fractal, self
Rendering of the butterfly by Hofstadter
Hofstadter's butterfly monochrome with axes en
Hofstadter 2002
Landau levels
Videos
In quantum mechanics, the energies of cyclotron orbits of charged particles in a uniform magnetic field are quantized to discrete values, thus known as Landau levels. These levels are degenerate, with
Hofstadter's butterfly
…Q are distinct prime numbers, there are exactly Q energy bands. For large Q ≫ P , the energy bands converge to thin energy bands corresponding to the Landau levels. Gregory Wannier showed that by taking into account the density of states, one can obtain a Diophantine equation that describes the system, as…
Diagram of a cyclotron orbit of a particle with speed v, which is the classical trajectory of a charged particle (here positive charge) under a uniform magnetic field B. The Landau quantization refers to a quantum charged particle under a uniform magnetic field.