Wavepacket phase-space quantum Monte Carlo method
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Wavepacket phase‑space quantum Monte Carlo method Ian Welland1 · David K. Ferry1 Received: 24 June 2020 / Accepted: 30 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We develop a novel numerical method for solving the Wigner equation which does not rely upon evaluation of the Wigner kernel and utilizes quantum trajectories. This method is a generalization of the effective potential approach widely used in semiconductor TCAD tools while also solving the full Wigner equation. The method’s accuracy is demonstrated with several standard examples. Keywords Wigner functions · Quantum trajectories · Device simulation
1 Introduction Modeling quantum effects in engineering, chemistry, and physics remains an active area of research on two fronts. The first of these is numerical accuracy, on which much progress has been made [1–3]. The latter of these is intuitive understanding, which often can be neglected in favor of standard formulations which limit introspection into quantum models. Significant interest has developed in the engineering of quantum devices which explicitly depend upon quantum effects [4] which presents the need for modeling approaches that allow for introspection into device function. Several properties of wave mechanics and matrix mechanics make these formulations abstract. The heart of the problem is the lack of trajectories in traditional quantum mechanics, forcing models to be built either in terms of operator observables or wavefunctions, the former of which is not really capable of visualization and the latter of which is not easily interpreted. However, in 1932, Wigner [5] developed the phase-space formulation of quantum mechanics. The intention was fairly innocuous: to find similar mathematical footing for quantum statistics as for classical statistics, to make it easier to compute quantum corrections to classical quantities. Wigner found a scalar function of the phase-space eigenvalues x and p which could be used in the same fashion as the classical * Ian Welland [email protected] 1
School of Electrical, Computer, and Energy Engineering, Arizona State University, Tempe, AZ 85287‑5706, USA
distribution function, namely it could be used to compute expectation values of quantum observables in exactly the same fashion as in classical statistics [6]:
⟨A(x, p)⟩ =
∫
dxdpA(x, p)fW (x, p).
(1)
This function fW , now called the Wigner function or phasespace function, has many other useful relations and properties. In particular, unlike the wavefunction or operators of wave and matrix mechanics, the Wigner function may be plotted as a phase-space function with definite interpretations, as it is entirely a real-valued and scalar function. The difficulty behind its use which has led to the Wigner function finding limited popularity outside of a few niche fields lies in its evolution equation. This equation bears considerable similarity to the classical equivalent, the Boltzmann equation in the absence of collisions:
𝜕f + v ⋅ ∇x f − qE ⋅ ∇k f = 0. 𝜕t
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