# Continuous Transition between Quantum and Classical Behavior for a Harmonic Oscillator

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.

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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.

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From the wavefunction in the eikonal representation

ψ=expω--it+(2x-cos(ωt))

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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.

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x(t)=cos(ωt)+

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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).