Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Random walks, continuum limits, and Schrödinger’s equation

Garnet N. Ord, A. S. Deakin

Published Nov 1, 1996• 38 citations
Publisher

Abstract

By considering the simple binary symmetric random walk on a discrete lattice in 1+1 dimensions, we show that the discrete analog of Schr\"odinger's equation describes a simple counting problem involving the sample paths on the lattice. Schr\"odinger's equation is obtained in the continuum limit with the result that this equation is confirmed to have a classical as well as a quantum context. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.3772

Related papers

Random walks and Schrödinger's equation in dimensions

Garnet N. Ord, A. S. Deakin

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.

Journal of Physics A Mathematical and General•Feb 7, 1997•16 citations

The Schrödinger and diffusion propagators coexisting on a lattice

Garnet N. Ord

The Schrödinger and diffusion equations are normally related only through a formal analytic continuation. There are apparently no intermediary partial differential equations with physical interpretations that can form a conceptual bridge between the two. However, if one starts off with a symmetric binary random walk on a lattice then it is possible to show that both equations occur as approximate descriptions of different aspects of the same classical probabilistic system. This suggests that lattice calculations may prove to be a useful intermediary between classical and quantum physics.

Journal of Physics A Mathematical and General•Mar 7, 1996•32 citations

Discrete spacetime, quantum walks, and relativistic wave equations

Leonard D. Mlodinow, Todd A. Brun

It has been observed that quantum walks on regular lattices can give rise to wave equations for relativistic particles in the continuum limit. In this paper, we define the three-dimensional discrete-time walk as a product of three coined one-dimensional walks. The factor corresponding to each one-dimensional walk involves two projection operators that act on an internal coin space; each projector is associated with either the ``forward'' or ``backward'' direction in that physical dimension. We show that the simple requirement that there is no preferred axis or direction along an axis---that is, that the walk be symmetric under parity transformations and steps along different axes of the cubic lattice be uncorrelated---leads, in the case of the simplest solution, to the requirement that the continuum limit of the walk is fully Lorentz-invariant. We show further that, in the case of a massive particle, this symmetry requirement necessitates the use of a four-dimensional internal space (as in the Dirac equation). The ``coin flip'' operation is generated by the parity transformation on the internal coin space, while the differences of the projection operators associated with each dimension must all anticommute. Finally, we discuss the leading correction to the continuum limit, and the possibility of distinguishing through experiment between the discrete random walk and the continuum-based Dirac equation as a description of fermion dynamics.

arXiv:1802.03910•Physical Review A•Apr 30, 2018•20 citations

Quantum equations from Brownian motion

B. S. Rajput

The Schrödinger free particle equation in 1+1 dimension describes second-order effects in ensembles of lattice random walks, in addition to its role in quantum mechanics, and its solutions represent the continuous limit of a property of ensembles of Brownian particles. In the present paper, the classical Schrödinger and Dirac equations have been derived from the Brownian motions of a particle, and it has been shown that the classical Schrödinger equation can be transformed into the usual Schrödinger quantum equation on applying the Heisenberg uncertainty principle between position and momentum, while the Dirac quantum equation follows from its classical counterpart on applying the Heisenberg uncertainty principle between energy and time, without applying any analytical continuation.

Canadian Journal of Physics•Feb 1, 2011•1 citation

Quantum walks, the discrete wave equation and Chebyshev polynomials

Simon Apers, Laurent Miclo

A quantum walk is the quantum analogue of a random walk. While it is relatively well understood how quantum walks can speed up random walk hitting times, it is a long-standing open question to what extent quantum walks can speed up the spreading or mixing rate of random walks on graphs. In this expository paper, inspired by a blog post by Terence Tao, we describe a particular perspective on this question that derives quantum walks from the discrete wave equation on graphs. This yields a description of the quantum walk dynamics as simply applying a Chebyshev polynomial to the random walk transition matrix. This perspective decouples the problem from its quantum origin, and highlights connections to earlier (non-quantum) work and the use of Chebyshev polynomials in random walk theory as in the Varopoulos-Carne bound. We illustrate the approach by proving a weak limit of the quantum walk dynamics on the lattice. This gives a different proof of the quadratically improved spreading behavior of quantum walks on lattices.

arXiv:2402.07809•Feb 12, 2024•0 citations

Schrödinger's Equation and Classical Brownian Motion

Garnet N. Ord

In the context of quantum mechanics, Schrödinger's equation has many possible interpretations, none of which appear to be entirely satisfactory. One reason for this is that there is no known microscopic model which yields both Schrödinger's equation and an accompanying ‘sensible’ interpretation. In contrast to this the diffusion equation is well known as a phenomenological equation obtainable from the microscopic model of Brownian motion. In this case the microscopic model provides an unambiguous interpretation of the partial differential equation. In this article we review work which shows that Schrödinger's equation by itself is also easily obtained from a lattice random walk version of the Brownian motion model. In this system, Schrödinger's equation arises by projection and the interpretation of wave functions is as direct and unambiguous as the interpretation of density in the diffusion case. However the direct interpretation is compatible with quantum mechanics only in a statistical sense in the context of ensembles of particles. This shows that measurement theory is the aspect of quantum theory which harbours the problems of interpretation, and we suggest the study of this and related systems to clearly illuminate the boundary between dynamics and measurement.

Fortschritte der Physik•Nov 1, 1998•5 citations
By considering the simple binary symmetric random walk on a discrete lattice in 1+1 dimensions, we show that the discrete analog of Schr"odinger's equation describes a simple counting problem involving the sample paths on the lattice. Schr"odinger's equation is obtained in the continuum limit with the result that this equation is confirmed to have a classical as well as a quantum context. \textcopyright{} 1996 The American Physical Society.
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.
The Schrödinger and diffusion equations are normally related only through a formal analytic continuation. There are apparently no intermediary partial differential equations with physical interpretations that can form a conceptual bridge between the two. However, if one starts off with a symmetric binary random walk on a lattice then it is possible to show that both equations occur as approximate descriptions of different aspects of the same classical probabilistic system. This suggests that lattice calculations may prove to be a useful intermediary between classical and quantum physics.
It has been observed that quantum walks on regular lattices can give rise to wave equations for relativistic particles in the continuum limit. In this paper, we define the three-dimensional discrete-time walk as a product of three coined one-dimensional walks. The factor corresponding to each one-dimensional walk involves two projection operators that act on an internal coin space; each projector is associated with either the ``forward'' or ``backward'' direction in that physical dimension. We show that the simple requirement that there is no preferred axis or direction along an axis---that is, that the walk be symmetric under parity transformations and steps along different axes of the cubic lattice be uncorrelated---leads, in the case of the simplest solution, to the requirement that the continuum limit of the walk is fully Lorentz-invariant. We show further that, in the case of a massive particle, this symmetry requirement necessitates the use of a four-dimensional internal space (as in the Dirac equation). The ``coin flip'' operation is generated by the parity transformation on the internal coin space, while the differences of the projection operators associated with each dimension must all anticommute. Finally, we discuss the leading correction to the continuum limit, and the possibility of distinguishing through experiment between the discrete random walk and the continuum-based Dirac equation as a description of fermion dynamics.
The Schrödinger free particle equation in 1+1 dimension describes second-order effects in ensembles of lattice random walks, in addition to its role in quantum mechanics, and its solutions represent the continuous limit of a property of ensembles of Brownian particles. In the present paper, the classical Schrödinger and Dirac equations have been derived from the Brownian motions of a particle, and it has been shown that the classical Schrödinger equation can be transformed into the usual Schrödinger quantum equation on applying the Heisenberg uncertainty principle between position and momentum, while the Dirac quantum equation follows from its classical counterpart on applying the Heisenberg uncertainty principle between energy and time, without applying any analytical continuation.
A quantum walk is the quantum analogue of a random walk. While it is relatively well understood how quantum walks can speed up random walk hitting times, it is a long-standing open question to what extent quantum walks can speed up the spreading or mixing rate of random walks on graphs. In this expository paper, inspired by a blog post by Terence Tao, we describe a particular perspective on this question that derives quantum walks from the discrete wave equation on graphs. This yields a description of the quantum walk dynamics as simply applying a Chebyshev polynomial to the random walk transition matrix. This perspective decouples the problem from its quantum origin, and highlights connections to earlier (non-quantum) work and the use of Chebyshev polynomials in random walk theory as in the Varopoulos-Carne bound. We illustrate the approach by proving a weak limit of the quantum walk dynamics on the lattice. This gives a different proof of the quadratically improved spreading behavior of quantum walks on lattices.
In the context of quantum mechanics, Schrödinger's equation has many possible interpretations, none of which appear to be entirely satisfactory. One reason for this is that there is no known microscopic model which yields both Schrödinger's equation and an accompanying ‘sensible’ interpretation. In contrast to this the diffusion equation is well known as a phenomenological equation obtainable from the microscopic model of Brownian motion. In this case the microscopic model provides an unambiguous interpretation of the partial differential equation. In this article we review work which shows that Schrödinger's equation by itself is also easily obtained from a lattice random walk version of the Brownian motion model. In this system, Schrödinger's equation arises by projection and the interpretation of wave functions is as direct and unambiguous as the interpretation of density in the diffusion case. However the direct interpretation is compatible with quantum mechanics only in a statistical sense in the context of ensembles of particles. This shows that measurement theory is the aspect of quantum theory which harbours the problems of interpretation, and we suggest the study of this and related systems to clearly illuminate the boundary between dynamics and measurement.