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Asymptotic Momentum of Dirac Particles in One Space Dimension

Kabir Narayanan, Abigail Perryman, A. Shadi Tahvildar-Zadeh

TL;DR

The study addresses how a free Dirac particle in one dimension exhibits asymptotic plane-wave behavior in a Bohmian framework. By employing a macroscopic scaling and a rigorous stationary phase analysis, the authors show the wave function decomposes into two counter-propagating packets with energies ±E and fixed momentum k, whose separation yields straight-line trajectories with constant velocity. The sign of the asymptotic energy is determined by the initial position, resolving how negative-energy components influence motion. This work provides a concrete link between Bohmian trajectories and Dirac-plane-wave dispersion, offering a microscopic justification for using plane-wave notions in scattering contexts like Compton scattering, with future directions including massless particles and outgoing-particle entanglement considerations.

Abstract

We analyze the trajectories of a massive particle in one space dimension whose motion is guided by a spin-half wave function that evolves according to the free Dirac equation, with its initial wave function being a Gaussian wave packet with a nonzero expected value of momentum $k$ and the positive expected value of energy $E = \sqrt{m^2+k^2}$. We prove that at large times, the wave function becomes {\em locally} a plane wave, which corresponds to trajectories with fixed values for asymptotic momentum $k$ and asymptotic energy $E$ or $-E$. 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.

Asymptotic Momentum of Dirac Particles in One Space Dimension

TL;DR

The study addresses how a free Dirac particle in one dimension exhibits asymptotic plane-wave behavior in a Bohmian framework. By employing a macroscopic scaling and a rigorous stationary phase analysis, the authors show the wave function decomposes into two counter-propagating packets with energies ±E and fixed momentum k, whose separation yields straight-line trajectories with constant velocity. The sign of the asymptotic energy is determined by the initial position, resolving how negative-energy components influence motion. This work provides a concrete link between Bohmian trajectories and Dirac-plane-wave dispersion, offering a microscopic justification for using plane-wave notions in scattering contexts like Compton scattering, with future directions including massless particles and outgoing-particle entanglement considerations.

Abstract

We analyze the trajectories of a massive particle in one space dimension whose motion is guided by a spin-half wave function that evolves according to the free Dirac equation, with its initial wave function being a Gaussian wave packet with a nonzero expected value of momentum and the positive expected value of energy . We prove that at large times, the wave function becomes {\em locally} a plane wave, which corresponds to trajectories with fixed values for asymptotic momentum and asymptotic energy or . 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.
Paper Structure (14 sections, 5 theorems, 138 equations, 5 figures)

This paper contains 14 sections, 5 theorems, 138 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Nonrelativistic particles
  3. Free Dirac Particles
  4. Cayley-Klein parameters and the Bloch sphere representation
  5. Classical and Quantum notions of Energy and Momentum
  6. Spinor Wave Packets
  7. Expected Values for Wave Packets
  8. Classical Limit of the Dirac Equation
  9. Solution of the Dirac equation in macroscopic variables
  10. The Stationary Phase Approximation
  11. Analysis of Trajectories
  12. Summary and Outlook
  13. Proof of Stationary Phase Approximation Theorem
  14. Numerical Investigations

Key Result

Theorem 2.1

Given initial data of the form with momentum $k \in \mathbb{R}$, $k \ne 0$, and energy $E = \sqrt{m^2 + k^2}$, the solution to the initial value problem for the Dirac equation (eq:DirEl, initdat) as $t\to \infty$ asymptotically approaches the superposition of two wave packets with energies $E$ and $-E$, both with momentum $k$, th

Figures (5)

  • Figure 1: The curves $y=y_0(x)$ and $y = B_\pm(x)$ for $\Theta_0 = \frac{\pi}{8}$ (on the left) and $\Theta_0 = \frac{3\pi}{8}$ (on the right).
  • Figure 2: Barrier construction in the case $\Theta_0<\pi/4$
  • Figure 3: 50 electron trajectories. Each row demonstrates the effect of varying one of the initial parameters $\Theta_0$, $k_0$, $m_{\text{el}}$, and $\sigma$.
  • Figure 4: The Bloch sphere representation of a single electron trajectory over 8 seconds (left) vs. 100 trajectories after 8 seconds (right).
  • Figure 5: Numerical vs. predicted values of Bloch variables for 50 trajectories after 6 seconds, for $k_0 = 10$, $m_{\text{el}} = 3$, $\sigma = 1$, $\Theta_0 = \pi/2$.

Theorems & Definitions (10)

  • Remark 1
  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • Corollary 3.2.1
  • Definition 3.3
  • Lemma 3.4
  • proof
  • proof