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 picture line above is the Heisenberg Hamiltonian "H"("t"), which may differ from the Schrödinger Hamiltonian. An important special case of the equation above is obtained if the Hamiltonian does not vary with time. Then the time-evolution operator can be written as Therefore, and, Here ∂"A"/∂"t" is the time derivative of the initial "A", not the "A"("t") operator defined. The last equation holds since commutes with . If the Hamiltonian does not vary with time, then the time-evolution operator can be written as where is the Hamiltonian and is the reduced Planck constant. Therefore, Thus and hence emerges the above Heisenberg equation