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Concept# Heisenberg picture

Summary

In physics, the Heisenberg picture or Heisenberg representation is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, but the state vectors are time-independent, an arbitrary fixed basis rigidly underlying the theory.
It stands in contrast to the Schrödinger picture in which the operators are constant, instead, and the states evolve in time. The two pictures only differ by a basis change with respect to time-dependency, which corresponds to the difference between active and passive transformations. The Heisenberg picture is the formulation of matrix mechanics in an arbitrary basis, in which the Hamiltonian is not necessarily diagonal.
It further serves to define a third, hybrid, picture, the interaction picture.
Mathematical details
In the Heisenberg picture of quantum mechanics the state vectors |ψ⟩ do not change with time, while observables A satisfy

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Frédéric Mila, Pierre Marcel Nataf, Kianna Wan

We present a method for the exact diagonalization of the SU(N) Heisenberg interaction Hamiltonian using Young tableaus to work directly in each irreducible representation of the global SU(N) group. This generalized scheme is applicable to chains consisting of several particles per site, with any SU(N) symmetry at each site. Extending some of the key results of substitutional analysis, we demonstrate how basis states can be efficiently constructed for the relevant SU(N) subsector, which, especially with increasing values of N or numbers of sites, has a much smaller dimension than the full Hilbert space. This allows us to analyze systems of larger sizes than can be handled by existing techniques. We apply this method to investigate the presence of edge states in SU(N) Heisenberg and Affleck-Kennedy-Lieb-Tasaki Hamiltonians.

2017Neutron inelastic scattering has been used to measure the magnetic excitations in powdered NiPS3, a quasitwo-dimensional antiferromagnet with spin S = 1 on a honeycomb lattice. The spectra show clear, dispersive magnons with a similar to 7 meV gap at the Brillouin zone center. The data were fitted using a Heisenberg Hamiltonian with a single-ion anisotropy assuming no magnetic exchange between the honeycomb planes. Magnetic exchange interactions up to the third intraplanar nearest neighbor were required. The fits show robustly that NiPS 3 has an easy-axis anisotropy with Delta = 0.3 meV and that the third nearest neighbor has a strong antiferromagnetic exchange of J(3) = -6.90 meV. The data can be fitted reasonably well with either J(1) < 0 or J(1) > 0, however, the best quantitative agreement with high-resolution data indicates that the nearest-neighbor interaction is ferromagnetic with J(1) = 1.9 meV and that the second nearest-neighbor exchange is small and antiferromagnetic with J(2) = -0.1 meV. The dispersion has a minimum in the Brillouin zone corner that is slightly larger than that at the Brillouin zone center, indicating that the magnetic structure of NiPS3 is close to being unstable.

Denys Osypenko, Shriya Palchaudhuri, Ralf Schneggenburger

Unraveling the neuronal mechanisms of fear learning might allow neuroscientists to make links between a learned behavior and the underlying plasticity at specific synaptic connections. In fear learning, an innocuous sensory event such as a tone (called the conditioned stimulus, CS) acquires an emotional value when paired with an aversive outcome (unconditioned stimulus, US). Here, we review earlier studies that have shown that synaptic plasticity at thalamic and cortical afferents to the lateral amygdala (LA) is critical for the formation of auditory-cued fear memories. Despite the early progress, it has remained unclear whether there are separate synaptic inputs that carry US information to the LA to act as a teaching signal for plasticity at CS-coding synapses. Recent findings have begun to fill this gap by showing, first, that thalamic and cortical auditory afferents can also carry US information; second, that the release of neuromodulators contributes to US-driven teaching signals; and third, that synaptic plasticity additionally happens at connections up- and downstream of the LA. Together, a picture emerges in which coordinated synaptic plasticity in serial and parallel circuits enables the formation of a finely regulated fear memory.