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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]
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 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 ...
Uncertainty principle of Heisenberg, 1927. The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known. In other words, the ...
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 .
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).
(It would mean that it had simultaneously zero momentum and well-defined position, which contradicts Heisenberg's uncertainty principle). Since the nuclear positions are constants, the electronic kinetic energy operator is invariant under translation over any nuclear vector.
The Bohr model was based on the assumed quantization of angular momentum according to = =. According to de Broglie, the electron is described by a wave, and a whole number of wavelengths must fit along the circumference of the electron's orbit: n λ = 2 π r . {\displaystyle n\lambda =2\pi r.}