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arXiv:quant-ph/0301156

Wave-packet trains of a time-dependent harmonic oscillator

Wenhua Hai, Shengxun Huang, Kelin Gao

Published Jan 29, 2003• 0 citations
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Abstract

By using a test-function method, we construct $n$ exact solutions of a quantum harmonic oscillator with a time-dependent "spring constant". Any $n$-th solution describes a wave-packet train consisting of $n+1$ packets. Its center oscillates like the classically harmonic oscillator with variable frequency, and width and highness of each packet change simultaneously. When the deformation is small, it behaves like a soliton train, and the large deformation is identified with collapse and revival of the wave-packet train.

DOI
10.48550/arxiv.quant-ph/0301156

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By using a test-function method, we construct nnn exact solutions of a quantum harmonic oscillator with a time-dependent "spring constant". Any nnn-th solution describes a wave-packet train consisting of n+1n+1n+1 packets. Its center oscillates like the classically harmonic oscillator with variable frequency, and width and highness of each packet change simultaneously. When the deformation is small, it behaves like a soliton train, and the large deformation is identified with collapse and revival of the wave-packet train.
By solving exactly the time dependent Schrdinger equation for simple harmonic oscillator, we have obtained respectively the wave packet solutions associated with the displaced ground state and the displaced excited eigenstates, and then demonstrated the centroid of the wave packet oscillates according to the classical law and that the wave packet form remains unchanged. For simple harmonic oscillator with the force constant depending on time the wave packets with the same structures as that of the usual simple harmonic oscillator can still be found to move in such a way that the centroids move according to the classical law, but their wave packets may deform in time. Depending on how the force constant varies with time, the wave packets can spread, contract or pulsate.
We find a set of different orthonormalized states of a nonstationary harmonic oscillator and use them to expand the solution of the Gross-Pitaevskii equation with harmonic potential. The expansion series describes wave-packet trains of a Bose-Einstein condensate, which may be induced initially by the modulational instability. The center of any wave-packet train oscillates like a classical harmonic oscillator of frequency $\ensuremath{\omega}$. The width and height of the wave packet and the distance between two wave packets change simultaneously like an array of breathers with frequency $2\ensuremath{\omega}$. We demonstrate analytically and numerically that for a set of suitable parameters the wave-packet trains can be more exactly fitted to the matter-wave soliton trains observed by Strecker et al. and reported in Nature (London) 417, 150 (2002).
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We show that one can obtain analytic solutions of the time-dependent Schrödinger equation that are more complex than the well-known oscillating coherent wave packet. Such Hermite-Gaussian or initially square wave packets exist for a free particle or for one subject to the harmonic oscillator potential. In either case, the Hermite-Gaussian packets retain their nodal structure even after long times. There is a single class of exact solutions for the system with oscillator constant K > 0, K = 0, or K < 0, leading to wave functions for the harmonic oscillator, the free particle, and the inverted oscillator, respectively.