The Schrödinger and diffusion propagators coexisting on a lattice

Ord, G. N.

doi:10.1088/0305-4470/29/5/007

Cited by 33 publications

(27 citation statements)

References 4 publications

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“…Although they locally propagate as particles, and then, are scattered as waves, the currents cannot be interpreted either as fields or as particles. This is also in contrast with other approaches where the Schrödinger and Maxwell equations have been obtained from the dynamics of ensembles of Brownian particles [17][18][19], thus providing a particle picture to standard wave equations.…”

Section: Discussionmentioning

confidence: 97%

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A New Construction for Scalar Wave Equations in Inhomogeneous Media

Arias

Vanneste

1997

J. Phys. I France

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The paper describes a formulation of discrete scalar wave propagation in an inhomogeneous medium by the use of elementary processes obeying a discrete Huygens' principle and satisfying fundamental symmetries such as time-reversal, reciprocity and isotropy. Its novelty is the systematic derivation of a unified equation which, properly tuned by a single parameter, leads to either the Klein-Gordon equation or the Schrödinger equation. The generality of this method enables one to consider its extension to other types of discrete wave equations on any kind of discrete lattice

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Section: Discussionmentioning

confidence: 97%

“…For this purpose, we introduce the ratio ν = λ 2d+1 /λ 1 . From the second equation of (17) and (14), we write…”

Section: Isotropymentioning

confidence: 99%

A New Construction for Scalar Wave Equations in Inhomogeneous Media

Arias

Vanneste

1997

J. Phys. I France

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“…Research topics include: exact solutions to the Cauchy problem, asymptotic behavior, estimates, related spatio-temporal structures, wave traveling, boundary conditions, etc. Some of the recent papers related to solution of the Cauchy problem for the Scrödinger equation are [19,20,21,23,24,27,22,6,25,26], see also [45].…”

Section: Schrödinger Equation and Quantum Mechanicsmentioning

confidence: 99%

Formulas that Represent Cauchy Problem Solution for Momentum and Position Schrödinger Equation

Remizov

2018

Potential Anal

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In the paper we derive two formulas representing solutions of Cauchy problem for two Schrödinger equations: one-dimensional momentum space equation with polynomial potential, and multidimensional position space equation with locally square integrable potential. The first equation is a constant coefficients particular case of an evolution equation with derivatives of arbitrary high order and variable coefficients that do not change over time, this general equation is solved in the paper. We construct a family of translation operators in the space of square integrable functions and then use methods of functional analysis based on Chernoff product formula to prove that this family approximates the solution-giving semigroup. This leads us to some formulas that express the solution for Cauchy problem in terms of initial condition and coefficients of the equations studied.

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“…Previous studies have been study the quantization of the one dimensional oscillator with friction [3,4]. More recently, the problem of quantizing dissipative systems have been studied in [5,6].…”

Section: Introductionmentioning

confidence: 99%