Heisenberg model (quantum) Heisenberg model (quantum) The Heisenberg model is a statistical mechanical model used in the study of critical points and phase transitions of magnetic systems, in which the spins of the magnetic systems are treated quantum mechanically. In the prototypical Ising model, defined on a "d"-dimensional lattice, at each lattice site, a spin formula_1 represents a microscopic magnetic dipole to which the magnetic moment is either up or down. Except the coupling between magnetic dipole moments, there is also a multipolar version of Heisenberg model called the multipolar exchange interaction. For quantum mechanical reasons (see exchange interaction or the subchapter "quantum-mechanical
Heisenberg group of differential operators on ℝ with polynomial coefficients, since any such operator has a unique representation in the form This algebra is called the Weyl algebra. It follows from abstract nonsense that the Weyl algebra "W" is a quotient of formula_79. However, this is also easy to see directly from the above representations; viz. by the mapping The application that led Hermann Weyl to an explicit realization of the Heisenberg group was the question of why the Schrödinger picture and Heisenberg picture are physically equivalent. Abstractly, the reason is the Stone–von Neumann theorem: there is a unique unitary representation with