Random walks, continuum limits, and Schrödinger’s equation

Ord, G. N.; Deakin, A. S.

doi:10.1103/physreva.54.3772

Cited by 34 publications

(24 citation statements)

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“…Thus, for example, if we start all the particles off in state one, then after one step roughly half of them are in state one and half in state two (assuming that ar1Â2). It then takes eight time steps for the ensemble to return to its initial statistical state in the sense that the expected number of direction changes per particle is then 4 [2]. As a result the equation which we shall consider in the continuum limit is Eq.…”

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Schrödinger's Equation and Discrete Random Walks in a Potential Field

Ord

1996

Annals of Physics

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“…(13), (12) into Eq. (9), and keep only terms up to order ($) 2 . After some algebra, the eight power of the shift matrix on the right in Eq.…”

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Schrödinger's Equation and Discrete Random Walks in a Potential Field

Ord

1996

Annals of Physics

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“…More or less analogous random walks have already been introduced in the literature (see, e.g., [3,19,28,29]), but in another context.…”

Section: Remarkmentioning

confidence: 99%

“…(iii) Renvoyons pour la troisième aux fameux échiqier (chessboard) de Feynman [14], aux travaux d'Ord (cf. [28,29]) et à [3,19].…”

Section: Introductionunclassified

Probabilités et fluctuations quantiques

Fliess

2007

Comptes Rendus. Mathématique

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Reçu le 30 juin 2006 ; accepté après révision le 4 avril 2007 Disponible sur Internet le 11 mai 2007 Présenté par Yves Meyer RésuméCette Note esquisse une construction mathématique simple et naturelle du caractère probabiliste de la mécanique quantique. Elle utilise l'analyse non standard et repose sur l'interprétation due à Feynman, mise en avant dans certaines approches fractales, du principe d'incertitude de Heisenberg, c'est-à-dire des fluctuations quantiques. On aboutit ainsi à des équations différentielles stochastiques, comme dans la mécanique stochastique de Nelson, découlant de marches aléatoires infinitésimales. Pour citer cet article : M. Fliess, C. R. Acad. Sci. Paris, Ser. I 344 (2007). AbstractProbabilities and quantum fluctuations. This Note is sketching a simple and natural mathematical construction for explaining the probabilistic nature of quantum mechanics. It employs nonstandard analysis and is based on Feynman's interpretation of the Heisenberg uncertainty principle, i.e., of the quantum fluctuations, which was brought to the forefront in some fractal approaches. It results, as in Nelson's stochastic mechanics, in stochastic differential equations which are deduced from infinitesimal random walks. To cite this article: M. Fliess, C. R. Acad. Sci. Paris, Ser. I 344 (2007). Abridged English versionWe are sketching a 'simple and natural' explanation of the probabilistic nature of quantum mechanics. This is achieved by utilizing the mathematical formalism of nonstandard analysis and Feynman's interpretation of the Heisenberg uncertainty principle, i.e., of the quantum fluctuations. Nontechnical presentation of the main ideasThis summary is intended for readers who are not familiar with nonstandard analysis. As often in nonstandard analysis (see, e.g., [1,26]), we replace a continuous time interval by an infinite 'discrete' set of infinitely closed time instants. Substituting m x t for p in the well known expression of the uncertainty principle, where x is the position, m the mass, p the momentum, h = 2πh the Planck constant, yields Eq. (1). We rewrite it by postulating that the Adresse quantity (2), where δt > 0 is a given infinitesimal, is limited and appreciable, i.e., it is neither infinitely large nor infinitely small. Those computations are stemming from Feynman's interpretation [13,14] of the uncertainty principle (see, also, [21,27]): The 'quantum trajectories' are fractal curves, of Hausdorff dimension 2. 'Weak' mathematical assumptions permit to derive the infinitesimal difference equation (3). The lack of any further physical assumption yields the equiprobability of +1 and −1. If x is Markov, i.e., if b and σ are functions of t and x(t), and not of {x(τ ) | 0 τ < t}, the corresponding infinitesimal random walk is 'equivalent' to a stochastic differential equation in the usual sense (see, e.g., [1,5]). Remark 1.More or less analogous random walks have already been introduced in the literature (see, e.g., [3,19,28,29]), but in another context. Nonstandard analysisReplace the interval [0, ...

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“…There are now several such systems available, and in these models the objects of study are ensembles of classical particles where the systems exhibit quantum dynamics as secondorder effects. The nonrelativistic free particle in 1 + 1 dimensions is considered in (Ord, 1996a) and Ord and Deakin (1996), and a 16-state (2 + 1)-dimensional model may be found in Ord and Deakin (1997). Relativistic versions may be found in Ord (1996b, c).…”

Section: Introductionmentioning

confidence: 99%

Classical spin in a potential field

Ord

Deakin

1997

Int J Theor Phys

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We consider an ensemble of restricted discrete random walks in 2 + I dimensions. The restriction on the walks is such as to give particles an intrinsic angular momentum. The walks are embedded in a field which affects the mean free path of the walks. We show that the dynamics of the walks is such that second-order effects are described by a discrete form of Schr6dinger's equation for particles in a potential field. This provides a classical context of the equation which is independent of its quantum context.

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Random walks, continuum limits, and Schrödinger’s equation

Cited by 34 publications

References 9 publications

Schrödinger's Equation and Discrete Random Walks in a Potential Field

Schrödinger's Equation and Discrete Random Walks in a Potential Field

Probabilités et fluctuations quantiques

Classical spin in a potential field

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