Propagation of a relativistic electron wave packet in the Dirac equation

Solving the Dirac equation is a formidable task due to the high frequency and the degrees of freedom involved. However, this high frequency allows one to obtain an approximation to the equation. Here, we directly solve the Dirac equation using an envelope method and derive analytical solutions of Dirac wave packets to first order for the small momentum spread. We apply the insight gained from this solution to the Zitterbewegung behavior in a Dirac-like system, where we show that it is crucial to include the first-order term in our solution to correctly describe a Dirac packet.

Figure 1

(a) Dependences of positive-energy Dirac spinor coefficients on the momentum. (b) Three-dimensional visualization of spinor vectors of positive and negative energies as a function of momentum.

Figure 2

(a) Density plot of the probability and (b) line plot of the position expectation value of a total $(1+1)$ mixed wave function ($\Theta =\pi /4$) with $p_0=0.967mc$ and $2{\sigma }_p=0.25mc$. Red (light gray) dashed, blue (dark gray dashed), and gray solid lines indicate the position expectation values of positive-energy-component-only, negative-energy-component-only, and total mixed wave packets, respectively.

Figure 3

Comparison of the position expectation values from the analytical solution (red solid line) and the numerical simulations of the exact mixed state (blue dashed line) and the spinor state (green dotted line) at $\Theta =\pi /4$, with $p_0$ and ${\sigma }_p$ of (a) 0, 0.125, (b) 0, 0.25, (c) 0, 0.5, (d) 0.5, 0.125, (e) 0.5, 0.25, and (f) 0.5, 0.5 in units of $mc$. Note that spinor packets show a drift due to momentum shift.

Figure 4

Position expectation values of the total mixed wave function ($\Theta =0.5076$) from the analytical solution (red solid line) and the numerical simulations of the exact mixed state (blue dashed line) and the $\left(1,1\right)$ spinor state (green dotted line), where ${\sigma }_z=\Delta =\frac{{{\lambda }^{-}}_C}{0.62}$ and $k_0=\frac{1}{\Delta }$. Also plotted are the experimental (black squares) and simulated (black center line) mean positions in Ref. [13] (digitized from Fig. 2 therein).

Figure 5

(a) The initial angular frequencies, (b) the amplitudes, and (c) the maximum velocities of the Zitterbewegung behavior of the analytical solution (red solid line), the numerical simulations with exact Gaussian packets (black dashed line), and spinor wave packets (green dotted line) as a function of momentum spread at $p_0=0$. Also plotted are the experimental values (black squares) from Ref. [13] (digitized from their Fig. 1 inset).

Figure 6

Momentum profiles of total (black dashed line), positive-energy [red (light gray) solid line], and negative-energy [blue (dark gray) solid line] components of spinor-basis wave packets prepared with ${\sigma }_p=0.25mc$ in (a) $(1,0)$ and (b) $(1,1)$ spinors. Vertical lines indicate momentum shifts, calculated using Eqs. (60) and (61).

Figure 7

Deviation of position expectation values of the spinor-projected (solid line) and the energy-projected (dashed line) wave functions with $p_0=0.967mc$ (200 keV), $2{\sigma }_p=0.25mc$, and $\Theta =\pi /4$.