Quantum Phenomena Modeled by Interactions between Many Classical Worlds
Michael J. W. Hall Dirk-André Deckert and Howard M. Wiseman,
PHYSICAL REVIEW X 4, 041013 (23 October 2014)
Quantum Mechanics Without Wavefunctions
Jeremy Schiff and Bill Poirier
J. Chem. Phys. 136, 031102 (2012)
Comparison of Poirier's many-D expression to Hall's 1-D "toy" expression.
For Hall's finite number of interacting worlds we need the following difference operators instead of derivatives.
Schiff and Poirier eq. (18)
Compare to Hall's MIW quantum force
Create a numeric function that depends on the nearest neighbors:
x(n-2) x(n-1) x(n) x(n+1) x(n+2)
Why does Poirier eq. (20) include a second term?
Only the first term corresponds to Hall's expression.
Computation of total energy below shows that Hall's expression is correct.
Simulation of a single particle in one dimension over 100 worlds. The initial spacial distribution is a Gaussian "wave packet". The "uniformizing function" is the cummulative distribution, this case the error function. We can use the inverse of the uniformizing function to create a gaussian packet from a uniform range.
Total classical potential
Total quantum Potential
Total kinetic energy
Run the integration
Störmer–Verlet discretization with continuous integrating stepsize controller
Ref: Explicit, Time Reversible, Adaptive Step Size Control by Ernst Hairer and Gustaf Söderlind