Quantum particles are emitted by a source
, pass through two slits in a barrier and arrive at a screen. For simplicity, two identical Gaussian profiles are assumed with identical initial width
and group velocity
in

space, and the classical initial velocities are restricted to the initial velocities of the Bohmian trajectories. The group velocities
of the Gaussian profiles differ only in sign. For simplicity, we set the mass of the particle
, as well as
, equal to 1.
To see how the classical trajectories smoothly pass over to Bohmian trajectories, the general equation of motion is given by the acceleration term from the classical mechanics, which is the second derivative of the position
with respect to time
:
.
Here
is the quantum potential,
is the environment coupling function and
is defined by:
For
, the trajectories become classical because
reduces to 0. For
, the trajectories become Bohmian because
tends to 1, the quantum limit, which leads to the quantum motion. For increasing time, the amplitude of the quantum potential
decreases.
For more detailed information about the continuous transition between the classical and the Bohm trajectories, see [1–3] and for an animated example, see [4].
If you increase
PlotPoints,
AccuracyGoal,
PrecisionGoal and
MaxSteps, the results become more accurate. The starting positions of the particles are linearly distributed around the peaks of the wave density at
. The particle positions are plotted against their downscaled kinetic energy along the vertical axis. For certain time steps and for certain environment factors
(mesoscopic cases), the trajectories become very unstable.
The trajectories for the double slit were first numerically calculated in [5]. For more detailed information about Bohmian mechanics, see [6].
[1] P. Ghose, "A Continuous Transition between Quantum and Classical Mechanics. I,"
Foundations of Physics,
32(6), 2002 pp. 871–892.
doi:10.1023/A:1016055128428.
[2] P. Ghose and M. K. Samal, "A Continuous Transition between Quantum and Classical Mechanics. II,"
Foundations of Physics,
32(6), 2002 pp. 893–906.
doi:10.1023/A:1016007212498.
[4] P. Ghose and K. von Bloh.
Continuous Transitions between Quantum and Classical Motions for Young's Interference Experiment [Video]. (Sep 28, 2016)
www.youtube.com/watch?v=m_IHnJFSk0U.
[5] C. Philippidis, C. Dewdney and B. J. Hiley, "Quantum Interference and the Quantum Potential,"
Il Nuovo Cimento B,
52(1), 1979 pp. 15–28.
doi:10.1007/BF02743566.