Gauge-Invariant Semi-Discrete Wigner Theory
- URL: http://arxiv.org/abs/2208.09208v3
- Date: Wed, 23 Nov 2022 08:35:04 GMT
- Title: Gauge-Invariant Semi-Discrete Wigner Theory
- Authors: Mihail Nedjalkov, Mauro Ballicchia, Robert Kosik, Josef Weinbub
- Abstract summary: A gauge-invariant Wigner quantum mechanical theory is obtained by applying the Weyl-Stratonovich transform to the von Neumann equation for the density matrix.
We derive the evolution equation for the linear electromagnetic case and show that it significantly simplifies for a limit dictated by the long coherence length behavior.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: A gauge-invariant Wigner quantum mechanical theory is obtained by applying
the Weyl-Stratonovich transform to the von Neumann equation for the density
matrix. The transform reduces to the Weyl transform in the electrostatic limit,
when the vector potential and thus the magnetic field are zero. Both cases
involve a center-of-mass transform followed by a Fourier integral on the
relative coordinate introducing the momentum variable. The latter is continuous
if the limits of the integral are infinite or, equivalently, the coherence
length is infinite. However, the quantum theory involves Fourier transforms of
the electromagnetic field components, which imposes conditions on their
behavior at infinity. Conversely, quantum systems are bounded and often very
small, as is, for instance, the case in modern nanoelectronics. This implies a
finite coherence length, which avoids the need to regularize non-converging
Fourier integrals. Accordingly, the momentum space becomes discrete, giving
rise to momentum quantization and to a semi-discrete gauge-invariant Wigner
equation. To gain insights into the peculiarities of this theory one needs to
analyze the equation for specific electromagnetic conditions. We derive the
evolution equation for the linear electromagnetic case and show that it
significantly simplifies for a limit dictated by the long coherence length
behavior, which involves momentum derivatives. In the discrete momentum picture
these derivatives are presented by finite difference quantities which, together
with further approximations, allow to develop a computationally feasible model
that offers physical insights into the involved quantum processes. In
particular, a Fredholm integral equation of the second kind is obtained, where
the "power" of the kernel components, measuring their rate of modification of
the quantum evolution, can be evaluated.
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