The Wigner equation in the presence of electromagnetic potentials

Nedjalkov, Mihail; Weinbub, Josef; Ellinghaus, Paul; Selberherr, S.

doi:10.1007/s10825-015-0732-y

Cited by 7 publications

(8 citation statements)

References 15 publications

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“…This also indicates that a numerical incorporation of these terms should be in conjunction with their convolution structure as in the case of a gauge transform of the scalar potential Wigner equation [41]. Furthermore, such transform can be used to modify the Wigner potential in a way which is more convenient with respect to computational efficiency [41]. However, the numerical aspects of such a generalization are beyond the scope of this work: most importantly, the available signed particle approach is fully capable to simulate the interplay between constant magnetic and general spatially dependent electric fields posed by Eq.…”

Section: Computational Aspectsmentioning

confidence: 99%

“…From a classical point of view the particle evolution is governed by forces. Electromagnetic potentials are introduced as a mathematical construct, which simplifies the calculations and have no physical significance [41]. Quantum descriptions explicitly depend on the potentials via the Hamiltonian.…”

Section: Electromagnetic Potentials In the Wigner Picturementioning

confidence: 99%

“…The expression for D considerably simplifies in the case of a homogeneous magnetic field. For convenience, we consider (41). The first term in the square brackets completely disappears since B is a constant and the integration over τ can be performed directly.…”

Section: Homogeneous Electromagnetic Conditionsmentioning

confidence: 99%

“…This suggests that nonlocal effects due to the magnetic field are introduced if the latter is inhomogeneous, as it is in the case of the electric counterpart. This also indicates that a numerical incorporation of these terms should be in conjunction with their convolution structure as in the case of a gauge transform of the scalar potential Wigner equation [41]. Furthermore, such transform can be used to modify the Wigner potential in a way which is more convenient with respect to computational efficiency [41].…”

Section: Computational Aspectsmentioning

confidence: 99%

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Wigner equation for general electromagnetic fields: The Weyl-Stratonovich transform

et al. 2019

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Gauge-invariant Wigner theories are formulated in terms of the kinetic momentum, which-being a physical quantity-is conserved after a change of the gauge. These theories rely on a transform of the density matrix, originally introduced by Stratonovich, which generalizes the Weyl transform by involving the vector potential. We thus present an alternative derivation of the Weyl-Stratonovich transform, which bridges the concepts and notions used by the different, available gauge-invariant approaches and thus links physically intuitive with formal mathematical viewpoints. Furthermore, an explicit form of the Wigner equation, suitable for numerical analysis and corresponding to general, inhomogeneous, and time-dependent electromagnetic conditions, is obtained. For a constant magnetic field, the equation reduces to two models: in the case of a constant electric field, this is the ballistic Boltzmann equation, where classical particles are driven by local forces. The second model, derived for general electrostatic conditions, involves novel physics, where the magnetic field acts locally via the Liouville operator, while the electrostatics is determined by the manifestly nonlocal Wigner potential. A significant consequence of our work is the fact that now the constant magnetic field case can be treated with existing numerical approaches developed for the standard, scalar potential Wigner theory. Therefore, in order to demonstrate the feasibility of the approach, a stochastic method is applied to simulate a physically intuitive evolution problem.

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Section: Computational Aspectsmentioning

confidence: 99%

Section: Electromagnetic Potentials In the Wigner Picturementioning

confidence: 99%

Section: Homogeneous Electromagnetic Conditionsmentioning

confidence: 99%

Section: Computational Aspectsmentioning

confidence: 99%

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Wigner equation for general electromagnetic fields: The Weyl-Stratonovich transform

et al. 2019

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“…The evolution equation for a symplectic tomogram was obtained in [16]. On the other hand, the gauge properties known for the Schrödinger equation for the wave function and Moyal equation for the Wigner function [29] have not been considered until now in the tomographic representation of quantum mechanics, while gauge invariance of the Wigner-Moyal representation has been studied in sufficient detail [30][31][32][33].…”

Section: Introductionmentioning

confidence: 99%

Gauge transformation of quantum states in probability representation

Korennoy

Man’ko

2017

J. Phys. A: Math. Theor.

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The gauge invariance of the evolution equations of tomographic probability distribution functions of quantum particles in an electromagnetic field is illustrated. Explicit expressions for the transformations of ordinary tomograms of states under a gauge transformation of electromagnetic field potentials are obtained. Gauge-independent optical and symplectic tomographic quasi-distributions and tomographic probability distributions of states of quantum system are introduced, and their evolution equations having the Liouville equation in corresponding representations as the classical limit are found.

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Recent advances in Wigner function approaches

Weinbub

Ferry

2018

Applied Physics Reviews

237

174

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The Wigner function was formulated in 1932 by Eugene Paul Wigner, at a time when quantum mechanics was in its infancy. In doing so, he brought phase space representations into quantum mechanics. However, its unique nature also made it very interesting for classical approaches and for identifying the deviations from classical behavior and the entanglement that can occur in quantum systems. What stands out, though, is the feature to experimentally reconstruct the Wigner function, which provides far more information on the system than can be obtained by any other quantum approach. This feature is particularly important for the field of quantum information processing and quantum physics. However, the Wigner function finds wide-ranging use cases in other dominant and highly active fields as well, such as in quantum electronics—to model the electron transport, in quantum chemistry—to calculate the static and dynamical properties of many-body quantum systems, and in signal processing—to investigate waves passing through certain media. What is peculiar in recent years is a strong increase in applying it: Although originally formulated 86 years ago, only today the full potential of the Wigner function—both in ability and diversity—begins to surface. This review, as well as a growing, dedicated Wigner community, is a testament to this development and gives a broad and concise overview of recent advancements in different fields.

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The Wigner equation in the presence of electromagnetic potentials

Cited by 7 publications

References 15 publications

Wigner equation for general electromagnetic fields: The Weyl-Stratonovich transform

Wigner equation for general electromagnetic fields: The Weyl-Stratonovich transform

Gauge transformation of quantum states in probability representation

Recent advances in Wigner function approaches

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