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arXiv:0902.4355

Counterpropagating Wavepacket Solutions of the Time-Dependent Schroedinger Equation for a Decaying Potential Field

Babur M. Mirza

Published Feb 25, 2009• 0 citations
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Abstract

We investigate wavepacket solutions for time-dependent Schoedinger equation in the presence of an exponentially decaying potential. Assuming for travelling wave solutions the phase to be a linear combination of the space and time coordinates, we obtain two distinct wavepacket solutions for the Schroedinger equation. The wavepackets counterpropagate in space at a constant velocity without any distortion or spreading thus retain their initial form at arbitrarily large distances.

DOI
10.48550/arxiv.0902.4355

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We investigate wavepacket solutions for time-dependent Schoedinger equation in the presence of an exponentially decaying potential. Assuming for travelling wave solutions the phase to be a linear combination of the space and time coordinates, we obtain two distinct wavepacket solutions for the Schroedinger equation. The wavepackets counterpropagate in space at a constant velocity without any distortion or spreading thus retain their initial form at arbitrarily large distances.
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Solutions to the time-dependent Schr\"odinger equation for a particle in a spatially uniform time-dependent field and some potentials of arbitrary form are proved to be like the time-independent eigenfunctions for an identical static potential. The field phase modulates the wave function and translates its envelope. This work applies directly to constant, linear, and parabolic static potentials and is self-consistent. Results for the modulation by a time-dependent field of the current through a rectangular barrier are relevant to the traversal time for tunneling and associated device limitations.
The free Schrödinger equation has localized constant velocity solutions ψ v of the form ψ = f ( r − v t ) e iλ t with λ being a constant. These solutions are eigenvectors of a momentum operator such that . The wavepacket, while not normalizable, is both localized and in a definite momentum state. The ψ v are orthogonal in the inner product space ⟨ϕ| r 2 |ψ⟩, and the operator is symmetric therein. We discuss whether these ψ v can act as basis states rather than the usual plane waves.