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arXiv:quant-ph/0008077 · Journal of Physics A Mathematical and General

Diffraction of wavepackets in space and time

G Kälbermann

Published Aug 10, 2001• 12 citations
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Abstract

The phenomenon of wavepacket diffraction in space and time is described. It consists in a diffraction pattern whose spatial location progresses with time. The pattern is produced by wavepacket quantum scattering off an attractive or repulsive time-independent potential. An analytical formula for the pattern at t →∞ is derived both in one dimension and in three dimensions. The condition for the pattern to exist is developed. The phenomenon is shown numerically and analytically for the Dirac equation in one dimension also. An experiment for the verification of the phenomenon is described and simulated numerically.

DOI
10.1088/0305-4470/34/33/311

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The phenomenon of wavepacket diffraction in space and time is described. It consists in a diffraction pattern whose spatial location progresses with time. The pattern is produced by wavepacket quantum scattering off an attractive or repulsive time-independent potential. An analytical formula for the pattern at t →∞ is derived both in one dimension and in three dimensions. The condition for the pattern to exist is developed. The phenomenon is shown numerically and analytically for the Dirac equation in one dimension also. An experiment for the verification of the phenomenon is described and simulated numerically.
The phenomenon of wave packet diffraction in space and time is investigated numerically and analytically for a one-dimensional array of equally spaced finite-depth wells. Theoretical predictions for the lattice at long times and at low scattering energies coincide exactly with the results for a single well. At intermediate and short times compared to the classical passage time, the pattern shows both a broad diffractive pattern and an interference pattern inside each diffractive peak. The diffractive structure persists for this case to infinite time.
We study the quantum slit diffraction problem in three dimensions. In the treatment of diffraction of particles by a slit, it is usually assumed that the motion perpendicular to the slit is classical. Here we take into account the effect of the quantum nature of the motion perpendicular to the slit using the Green function approach [18]. We treat the diffraction of a Gaussian wave packet for general boundary conditions on the shutter. The difference between the standard and our three-dimensional slit diffraction models is analogous to the diffraction in time phenomenon introduced in [16]. We derive corrections to the standard formula for the diffraction pattern, and we point out situations in which this might be observable. In particular, we discuss the diffraction in space and time in the presence of gravity.