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Letters in Mathematical Physics

A simplex of probability measures associated with classical states of the harmonic oscillator

Alexander Bach, U. L�xmann-Ellinghaus

Published Feb 1, 1985• 4 citations
Publisher
DOI
10.1007/bf00400706

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The connection between the wavefunctions and the classical periodic orbits in a 2D harmonic oscillator is analytically constructed by using the representation of SU (2) coherent states. It is found that the constructed wavefunction generally corresponds to an ensemble of classical trajectories and its localization is extremely efficient. With the constructed wavefunction, we also analyse the property of the probability current density associated with the classical periodic orbit. The appearance of vortex structure in the quantum flow is clearly found to arise from the wave interference.
By solving exactly with a simple method the time dependent Schrödinger equation for simple harmonic oscillator, we demonstrate that the probability density associated with any displaced eigenstate has the following properties: (a) Its centroid oscillates according to the classical law. (b) Its wave packet form remains rigid. The same method is also applied to solve exactly the problem of simple harmonic oscillator with the force constant depending on time. In this case, probability densities with the same wave packet structures as that of the usual simple harmonic oscillator can be found to move in such a way that their centroids follow the classical law but their wave packets deform in time. Depending on how the force constant varies with time, the wave packets can spread, contract or pulsate.