To see how the Bohmian trajectories smoothly pass over to classical trajectories, the general equation of motion is given by the acceleration term from classical mechanics, which is the second derivative of the position

with respect to time

:

.

Here

is the quantum potential,

is the environmental coupling function, defined by:

with

and

.

To get the motion in the semi-classical regime, the acceleration term is numerically integrated, with initial velocities taken from the gradient of the phase for

, which is in this case zero.

For

, the trajectories become Bohmian because

tends to 1, the quantum limit, which leads to the quantum motion. For

, the trajectories become classical because

reduces to 0. The amplitude of the quantum potential

decreases with time.

For more detailed information about the continuous transition between classical and Bohm trajectories, see [1–3]. For an animated example, see [4]. For more detailed information about Bohmian mechanics, see [5].

The results become more accurate if you increase

PlotPoints,

AccuracyGoal,

PrecisionGoal and

MaxSteps. The starting positions of the particles are linearly distributed around the peak of the wave density at

.

[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 Motion for a Harmonic Oscillator Potential* [Video]. (Jan 5, 2017)

www.youtube.com/watch?v=EUvN8h-2KAQ.