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Garnet
N. Ord
Department of Math, Physics
and Computer Science
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Research Interests CoursesLinks
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Schroedinger's EquationThe Schroedinger and Diffusion Propagators Coexisting on a Lattice The Schroedinger 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. The above figure shows the appearance of the diffusive
and Feynman propagators at fixed time as the space-time lattice is refined.
Both these functions are observable characteristics of the same physical
system. (J. Phys. A. Lett. 7 March 1996)
Classical Spin in a Potential
Field
We consider an ensemble of restricted discrete random
walks in 2+1 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 are such that second order effects are described by a discrete
form of Schroedinger's equation for particles in a potential field. This
provides a classical context of the equation which is independent of its
quantum context.
Random Walks, Continuum Limits and Schroedinger's Equation By considering the simple binary symmetric random walk
on a discrete lattice in $1+1$ dimensions, we show that the discrete analog
of Schroedinger's equation describes a simple counting problem involving
the sample paths on the lattice. Schroedinger'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. (G.N. Ord and A.S. Deakin,
Phys. Rev. A v.54 #5 1996)
Schroedinger's Equation and Discrete Random Walks in a Potential Field It has been recently noted that the free particle Schroedinger
equation in 1+1 dimensions occurs naturally in the description of correlations
in random walks. In this non-quantum context, wave function solutions describe
features of ensembles of random walks on lattices and are as a consequence
observable and easily interpreted. In this article we extend this result
to show that the original random walk system may be modified to include
a smooth temperature field which locally alters the walker's expected mean
free path. This modification effects the correlations in such a way that
Schroedinger's free particle equation aquires a potential term. (G.N. Ord
Annals of Physics 250, August 1996)
Random Walks and Schroedinger's equation in 2+1 Dimensions We consider an ensemble of restricted discrete random
walks in 2+1 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 are such that second order effects are described by a discrete
form of Schroedinger'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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