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It is common to name the model depending on the values of , and : if , the model is called the Heisenberg XYZ model; in the case of = = =, it is the Heisenberg XXZ model; if = = =, it is the Heisenberg XXX model. The spin 1/2 Heisenberg model in one dimension may be solved exactly using the Bethe ansatz. [1]
where "H" and "S" label observables in Heisenberg and Schrödinger picture respectively, H is the Hamiltonian and [·,·] denotes the commutator of two operators (in this case H and A). Taking expectation values automatically yields the Ehrenfest theorem , featured in the correspondence principle .
The Heisenberg picture is closest to classical Hamiltonian mechanics (for example, the commutators appearing in the above equations directly correspond to classical Poisson brackets). The Schrödinger picture, the preferred formulation in introductory texts, is easy to visualize in terms of Hilbert space rotations of state vectors, although it ...
In physics, the Schrödinger picture or Schrödinger representation is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are mostly constant with respect to time (an exception is the Hamiltonian which may change if the potential changes).
The mathematical model of a channel used here is same as the classical one. Let : be a channel in the Heisenberg picture and : be a chosen ideal channel. To make the comparison possible, one needs to encode and decode Φ via appropriate devices, i.e. we consider the composition
Any possible choice of parts will yield a valid interaction picture; but in order for the interaction picture to be useful in simplifying the analysis of a problem, the parts will typically be chosen so that H 0,S is well understood and exactly solvable, while H 1,S contains some harder-to-analyze perturbation to this system.
The first model that was able to explain the full spectrum of thermal radiation was put forward by Max Planck in 1900. [9] He proposed a mathematical model in which the thermal radiation was in equilibrium with a set of harmonic oscillators. To reproduce the experimental results, he had to assume that each oscillator emitted an integer number ...
Heisenberg model can refer to two models in statistical mechanics: Heisenberg model (classical) , a classical nearest neighbour spin model Heisenberg model (quantum) , a model where the spins are treated quantum mechanically using Pauli matrices