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Journal of Mathematical Physics

Continuous-time random walk as a guide to fractional Schrödinger equation

Ervin Kaminski Lenzi, Haroldo Valentin Ribeiro, H. Mukai, R. S. Mendes

Published Sep 1, 2010• 20 citations
Publisher

Abstract

We argue that the continuous-time random walk approach may be a useful guide to extend the Schrödinger equation in order to incorporate nonlocal effects, avoiding the inconsistencies raised by Jeng et al. [J. Math. Phys. 51, 062102 (2010)]. As an application, we work out a free particle in a half space, obtaining the time dependent solution by considering an arbitrary initial condition.

DOI
10.1063/1.3491333

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We argue that the continuous-time random walk approach may be a useful guide to extend the Schrödinger equation in order to incorporate nonlocal effects, avoiding the inconsistencies raised by Jeng et al. [J. Math. Phys. 51, 062102 (2010)]. As an application, we work out a free particle in a half space, obtaining the time dependent solution by considering an arbitrary initial condition.
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Some recent results have shown that in addition to its role in quantum mechanics, the Schrödinger free particle equation in (1+1) dimensions describes second-order effects in ensembles of lattice random walks. This alternative classical context for Schrödinger's equation is independent of its role in quantum mechanics. In this paper we extend this result to include the case of Schrödinger's equation in (2+1) dimensions for a particle in a smooth bounded potential. The extension suggests that the new classical context of Schrödinger's equation is quite general.
Initially developed in the framework of quantum stochastic calculus, the main equations of quantum stochastic filtering were later on derived as the limits of Markov models of discrete measurements under appropriate scaling. In many branches of modern physics it became popular to extend random walk modeling to the continuous time random walk (CTRW) modeling, where the time between discrete events is taken to be non-exponential. In the present paper we apply the CTRW modeling to the continuous quantum measurements yielding the new fractional in time evolution equations of quantum filtering and thus new fractional equations of quantum mechanics of open systems. The related quantum control problems and games turn out to be described by the fractional Hamilton-Jacobi-Bellman (HJB) equations on Riemannian manifolds. By-passing we provide a full derivation of the standard quantum filtering equations, in a modified way as compared with existing texts, which (i) provides explicit rates of convergence (that are not available via the tightness of martingales approach developed previously) and (ii) allows for the direct applications of the basic results of CTRWs to deduce the final fractional filtering equations.
The position density of a "particle" performing a continuous-time quantum walk on the integer lattice, viewed on length scales inversely proportional to the time , converges (as tends to infinity) to a probability distribution that depends on the initial state of the particle. This convergence behavior has recently been demonstrated for the simplest continuous-time random walk [N. Konno, Phys. Rev. E 72, 026113 (2005)]. In this Brief Report, we use a different technique to establish the same convergence for a very large class of continuous-time quantum walks, and we identify the limit distribution in the general case.