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We prove that at large times, the wave function is approximately equal to the superposition of two wave packets traveling in opposite directions, which results in trajectories with approximately constant asymptotic momentum $k$ and asymptotic energy $\pm c^2\sqrt{m^2+k^2}$, with $m$ the rest mass of the particle and $c$ the speed of light. The sign of the asymptotic energy is determined by the initial position of the particle. Particles with negative energy will have an asymptotic velocity that is in the opposite direction of their momentum.
The proof uses the stationary phase approximation method, for which we establish a rigorous error bound.
From: A. Shadi Tahvildar-Zadeh [view email]
[v1]
Wed, 24 Dec 2025 21:08:25 UTC (2,402 KB)
[v2]
Tue, 16 Jun 2026 18:24:15 UTC (3,331 KB)
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