* 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*

Prof. John H. FrederickDr Clemens Woywod

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

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:

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

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 .

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

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

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

Scattering of 2D wavepacket from the Eckart barrierThe potential: Eckart barrier (a.u.): HeightWidthPosition2D Eckart barrier harmonic potential surface

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)

time: 108 auPropagation of the Gaussian wavepacket on the Eckart barrier-harmonic potential surface

time: 173 auPropagation of the Gaussian wavepacket on the Eckart barrier-harmonic potential surface

Scattering of 3D wavepacket from the Eckart barrier

Propagation of the Gaussian wavepacket on 3D Eckart barrier-harmonic potential surfacetime: 0 au

Propagation of the Gaussian wavepacket on 3D Eckart barrier-harmonic potential surfacetime: 108 au

Propagation of the Gaussian wavepacket on 3D Eckart barrier-harmonic potential surfacetime: 173 au

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