BibTeX
@MISC{04exactresults,
author = {},
title = {Exact Results for},
year = {2004}
}
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Abstract
We consider time-dependent Gaussian wave packet solutions of the Schrödinger equation (with arbitrary initial central position, x0, and momentum, p0, for an otherwise free particle, but with an infinite wall at x = 0, so-called bouncing wave packets.We showhowdifference ormirror solutions of the form(x, t) − (−x, t) can, in this case, be normalized exactly, allowing for the evaluation of a number of time-dependent expectation values and other quantities in closed form. For example, we calculate 〈p2〉t explicitly which illustrates how the free-particle kinetic (and hence total) energy is affected by the presence of the distant boundary. We also discuss the time dependence of the expectation values of position, 〈x〉t, and momentum, 〈p〉t, and their relation to the impulsive force during the ‘collision ’ with the wall. Finally, the x0,p0 → 0 limit is shown to reduce a special case of a non-standard free-particle Gaussian solution. The addition of this example to the literature then expands of the relatively small number of Gaussian solutions to quantum mechanical problems with familiar classical analogs (free particle, uniform acceleration, harmonic oscillator, unstable oscillator, and uniform magnetic field) available in closed form. 1.
Keyphrases
exact result closed form free particle impulsive force unstable oscillator time-dependent gaussian wave packet solution special case uniform magnetic field harmonic oscillator familiar classical analog distant boundary infinite wall free-particle kinetic non-standard free-particle gaussian solution time-dependent expectation value gaussian solution small number so-called bouncing wave packet ormirror solution uniform acceleration schro dinger equation arbitrary initial central position mechanical problem time dependence expectation value
