From Poirier’s Bohmian Mechanics without Wavefunctions to Hall’s Many Interacting Worlds (1-D)
From Poirier's Bohmian Mechanics without Wavefunctions
To Hall's Many Interacting Worlds#
Ref:
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)
http://arxiv.org/abs/1402.6144Quantum Mechanics Without Wavefunctions
Jeremy Schiff and Bill Poirier
J. Chem. Phys. 136, 031102 (2012)
http://arxiv.org/abs/1201.2382v1
1-D
Comparison of Poirier's many-D expression to Hall's 1-D "toy" expression.
Difference operators
For Hall's finite number of interacting worlds we need the following difference operators instead of derivatives.
etc.
Hall's "toy" expression
eqs. (24,25)
Poirier's expression
Quantum Force
Schiff and Poirier eq. (18)
Compare to Hall's MIW quantum force
Fast Numeric Functions
Create a numeric function that depends on the nearest neighbors: x(n-2) x(n-1) x(n) x(n+1) x(n+2)
Quantum Potential###
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.
Classical force
Classical potential
Simulation
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.
Initial conditions
Acceleration
Total classical potential
Total quantum Potential
Total kinetic energy
Solution
Run the integration
Velocity Verlet
Störmer–Verlet discretization with continuous integrating stepsize controller
Ref: Explicit, Time Reversible, Adaptive Step Size Control by Ernst Hairer and Gustaf Söderlind
Velocity
Final state