Continuous Transition between Quantum and Classical Behavior for a Harmonic Oscillator
This Demonstration explores the simple quantum harmonic oscillator to show a continuous transition between the quantum motion, as represented by Bohm trajectories, and classical behavior in - space. In chemistry and solid-state physics, the regime between microscopic and macroscopic scales is described as mesoscopic or semi-classical.
A nondispersive Gaussian wave packet can be constructed by a superposition of stationary eigenfunctions of the harmonic oscillator, in which the center of the packet oscillates harmonically between with frequency . The amplitude of the wave density is proportional to the square root of the frequency.
From the wavefunction in the eikonal representation
the gradient of the phase function and therefore the equation for the Bohmian trajectories can be calculated analytically. The motion is given by , where are the initial starting points. The trajectories of the particles oscillate with amplitude and frequency and they never cross. In the classical case, the situation is different. Over the ensemble of different initial positions, the particles oscillate over different centers and cross the classical amplitude In the semi-classical regime, the trajectories remain within the amplitude but can cross one another.
The graphic on the right shows the squared quantum wavefunction and the trajectories. The graphic on the left shows the particle positions, the squared quantum wavefunction (blue) and the quantum potential (red).
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 . For more detailed information about Bohmian mechanics, see .
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 .
 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.
 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.