Space-time evolution of Dirac wave packets

In this work we study the dynamics of free three-dimensional relativistic Gaussian wave packets with different spin polarizations. We analyze the connection between the symmetry of initial state and the dynamical characteristics of moving particle. The corresponding solutions of the Dirac equation having different types of symmetry were evaluated analytically and numerically and after that the electron probability densities, as well as the spin densities were visualized. The average values of velocity of the packet center and the average spin were calculated analytically, and the parameters of transient Zitterbewegung in different directions were obtained. These results can be useful for the interpretation of future experiments with trapped ions.

Figure 1

The electron probability density for the initial Gaussian packet, Eqs. (24) and (34): (a) at $z=0$, $k_0=0$, and $\Delta =5$, $d=1$ at time $t=7.5$; (b) at y $=$ 0, $k_0=0$, and $\Delta =5$, $d=5$ at time $t=7.5$.

Figure 2

The dependencies $W_{\pm }(p_z)$ of positive (solid line) and negative (dashed line) energy parts for the Gaussian initial wave packet, Eqs. (24) and (39) for $k_0=0$, $\Delta =5$, $d=1$ (a), and for $k_0=1$, $\Delta =5$, $d=5$ (b).

Figure 3

The electron probability density for the initial Gaussian packet, Eqs. (42) and (34): (a) at $z=0$ with $k_0=0$ and $\Delta =5$, $d=1$ at time $t=2\pi$; (b) at y $=$ 0, $k_0=1$ and $\Delta =5$, $d=2.5$ at time $t=7.5$.

Figure 4

The average projections of velocities ${\mathrm{V̄}}_x/c$ (a) and ${\mathrm{V̄}}_y/c$ (b) versus time $t$ (in units of ${\lambda }_k/c$) for the Gaussian initial wave packet, Eqs. (42) and (34) with $k_0=0$ and $\Delta =5$, $d=1$.

Figure 5

The average projection of velocity ${\mathrm{V̄}}_x/c$ (a) versus time $t$ (in units of ${\lambda }_k/c$) for the Gaussian initial wave packet, Eqs. (42) and (34) with $k_0=1$ and $\Delta =5$, $d=2.5$.

Figure 6

The distributions of the components of spin density ${\Sigma }_x(x,y,0,t)$, ${\Sigma }_y(x,y,0,t)$, ${\Sigma }_z(x,y,0,t)$ for the initial Gaussian packet, Eqs. (24) and (34) with $k_0=0$ and $\Delta =5$, $d=1$ at time $t=7.5$.

Figure 7

The distributions of the components of spin density ${\Sigma }_x(x,0,z,t)$, ${\Sigma }_z(x,0,z,t)$ for the initial Gaussian packet, Eqs. (42) and (34) with $k_0=0$ and $\Delta =d=3.64$ at time $t=8$.