Quantum Mechanics Without Wavefunctions
Jeremy Schiff and Bill Poirier
J. Chem. Phys. 136, 031102 (2012)
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)
Explicit, Time Reversible, Adaptive Step Size Control Ernst Hairer and Gustaf Söderlind SIAM Journal on Scientific Computing. 2005, vol. 26, no. 6, p. 1838-1851
Poirier’s many-D generalization, eq. (18), when specialized to 2-D and discretized by replacing derivatives with difference operators is the 2-D analog of Hall’s “toy” expression eq. (24). Let and represent the position of a particle (one particle in 2-D space) or the position of two particles (in 1-D configuration space) at an unspecified time in the world indexed by and .
For Hall's finite number of interacting worlds we will use the following difference operators instead of derivatives.
The Jacobian replacing derivatives with differences.
The Schiff and Poirier expression eq. (18) in 2-D has many more terms than in the 1-D case!
Create a numeric function that depends on just the nearest neighbors. This requires a function of 2x5x5 = 50 scalar quantities.
Why does Poirier eq. (20) include a second term? If we include it then total energy is not conserved!
Expand terms using symbolic determinants.
Manually reorganizing terms, we can put it into another form.
But for the computation we don't really need the determinants. (Only use the second term.)
Consider two particles in a one dimensional space over 4 worlds. The initial configuration space is shown below.
The particles approach each other with the same initial velocity in each world.
In order to compute the quantum force on the particles in world we introduce ficticious worlds at "approximate infinity" to represent open boundry conditions.
Run the integration using Störmer–Verlet discretization with a continuous integrating stepsize controller.
You might want something like this: https://addons.mozilla.org/En-US/firefox/addon/toggle-animated-gifs/ to manage the animation.
Start/stop GIF animations through a keyboard shortcut or by clicking them. You can also restart animations from the beginning, or disable animations by default. %md Each point in the following diagram represents the position of two particles. When the point approaches the x=y line
Each moving point in the following diagram represents the position of two particles in a single world (four worlds are shown). As the point approaches the x=y line (shown in red), the particles approach each other and the classical potential is at a maximum. In three worlds the particles overcome the potential barrier and continue. But one world (the last to approach) is reflected back. Although initially the two particles in each world approach each other with the same velocity, the particles in the last world to approach the barrier have lost energy to their counterparts in other worlds due to the quantum mechanical (interworld) interaction. When viewed in the projection we see two particles rebound instead of passing each other.