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* Currently at the Unilever Center for Molecular Informatics,
University of Cambridge Multidimensional de Broglie - Bohm
dynamics: the quantum motion with trajectoriesDr Dmitry Nerukh
University of Nevada, Reno*
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Prof. John H. FrederickDr Clemens Woywod
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de Broglie - Bohm interpretation of quantum mechanics
Quantizationparticle waveposition in spacefunctionobservables:
functions operators
Copenhagen interpretation of quantum mechanics
Einstein point of viewstatistical interpretation of wave
functionCausal interpretation of quantum mechanicspilot wave: Louis
de Broglie 1927 Solvay Conference quantum motion: David Bohm Phys.
Rev., 85, 166-179, 180-193 (1952)Peter R. Holland, The quantum
theory of motion, Cambridge University Press,
1993http://www.netcomuk.co.uk/~jvpearce/bohm/bohm.htmlhttp://www.tiac.net./users/fbh
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Hamilton-Jacobi theory Hamilton-Jacobi equation:
where - the Hamilton principal function,
single particle in external potential V:
,where
Hamilton-Jacobi equation:The equation of motion:
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generalized Hamilton-Jacobi theory:particle dynamicsevolution of
the S field define S as a physical field find particle
trajectories
the propagation of the S-functionparticle trajectorydefines the
particle trajectories; S has properties of a fieldGeneralized
Hamilton-Jacobi equationThe equation of motion
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Basic idea: quantum potential and quantum particlesThe
postulates of de Broglie-Bohm quantum theory of motiona system
comprises a wave propagating in space and time together with a
particle which moves continuously under the guidance of the
wave;the wave is described by , a solution to the Schrdinger
eq.;the particle motion is the solution to the equation
where is a phase of ; the initial condition must be specified;
the ensemble of the trajectories is generated by varying ;the
probability that a particle lies between and at time is given by
where .
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Reformulation of the Schrdinger equation
Quantum potential:
Wavefunction in a polar form=Substituting into the
time-dependent Schrdinger equationand separating real and imaginary
parts:compare to
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Simulation of a double slit experiment in two dimensions. The
trajectories of an ensemble at the lower slit only are superposed
on the interference pattern
Example
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Numerical Method
Delaunay tesselation of the argument points and mapping the line
along the derivative axis onto the tesselation structure. The
points of intersection of the upper red line and edges of the
triangles are used to approximate the cut of the surface
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Scattering of 2D wavepacket from the Eckart barrierThe
potential: Eckart barrier (a.u.): HeightWidthPosition2D Eckart
barrier harmonic potential surface
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time: 0 auPropagation of the Gaussian wavepacket on the Eckart
barrier-harmonic potential surfaceSimulation parameters: number of
particles: 386 initial kinetic energy of the particles: 20 a.u.time
step: 0.01 autotal propagation time: 1.2 aucalculation time: 0.5
hours (Pentium II, 400 MHz)
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time: 108 auPropagation of the Gaussian wavepacket on the Eckart
barrier-harmonic potential surface
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time: 173 auPropagation of the Gaussian wavepacket on the Eckart
barrier-harmonic potential surface
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Scattering of 3D wavepacket from the Eckart barrier
Propagation of the Gaussian wavepacket on 3D Eckart
barrier-harmonic potential surfacetime: 0 au
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Propagation of the Gaussian wavepacket on 3D Eckart
barrier-harmonic potential surfacetime: 108 au
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Propagation of the Gaussian wavepacket on 3D Eckart
barrier-harmonic potential surfacetime: 173 au
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Summary
It worksThe calculation time increases slower then exponential
with imensionalityExtremely well developed methods of propagating
classical trajectories and constructing Delaunay tesselationEasily
visualizableNatural introduction of various mixed quantum-classical
approaches