Figure 1

Wave packet falling on the lattice of $\delta$ barriers.Reuse & Permissions

Figure 2

Transmission coefficient spectrum $T=T\left(E\right)$ for the states of stationary scattering by the lattice of $N=15$ identical $\delta$ barriers of the power $\Omega =0.5\text{a.u.}$ disposed on the distance $d=5\mathrm{\AA }$ from each other.Reuse & Permissions

Figure 3

Probability density for the states of stationary scattering on the lattice of $N=15$ identical $\delta$ barriers corresponding to the spectral points were numerated in Fig. 2: (1), (3), (5), (6) near the resonance lines of total transparency, (2) between the lines, and (4) in the middle of transmission band.Reuse & Permissions

Figure 4

Scattering of the Gaussian wave packets: (a) the transmission coefficient spectrum $T=T\left(E\right)$ for the lattice composed of $N=50$ identical $\delta$ barriers with $\Omega =0.25\text{a.u.}$, $d=10\mathrm{\AA }$ in the region of $s=2,3,4$ transmission bands, and the group velocities of the free moving particle ${\upsilon }_{g0}$ and of the particle moving inside the lattice ${\upsilon }_g$ (black square in the band $s=3$ denotes the value ${\upsilon }_g=0.25\text{a.u.}$, measured by the slope of the line 3 in Fig. 11a, rhomb $({\upsilon }_g=0.43\text{a.u.})$, by slope of the line 3 in Fig. 11b; (b) spectral functions $\mid c\left(E\right)\mid$ for five Gaussian wave packets differing by their energetic position of maximums and by the width of their spectral functions. For the first of them $x_C=-500\mathrm{\AA }$, $\mathrm{\Delta }x=250\mathrm{\AA }$, $E_C={\hslash }^2{k}_C^2∕2m=3.35\mathrm{eV}$, for the second packet the values of $x_C$ and $\mathrm{\Delta }x$ are the same as for the first, but $E_C=2.55\mathrm{eV}$, for the third $x_C=-250\mathrm{\AA }$, $\mathrm{\Delta }x=45\mathrm{\AA }$, $E_C=3\mathrm{eV}$, for the fourth $x_C=-250\mathrm{\AA }$, $\mathrm{\Delta }x=60\mathrm{\AA }$, $E_C=3.35\mathrm{eV}$, and for the fifth $x_C=-250\mathrm{\AA }$, $\mathrm{\Delta }x=10\mathrm{\AA }$, $E_C=3.38\mathrm{eV}$.Reuse & Permissions

Figure 5

Initial stages of evolution of the spectrally narrow first wave packet from Fig. 4b with the spectral function covering two upper lines of the total transparence [the insertion in (a) sheet] to the transmitted impulse coming out and the quasistationary state forming. The dashed line depicts the initial packet location and dotted line the boundaries of lattice with length $L=Nd=500\mathrm{\AA }$; $\delta$ barriers are marked by the touches on the abscissa axis. The insertions show some details of the reflected and transmitted impulses in enlarged scale. The time is measured in atomic units.Reuse & Permissions

Figure 6

Next stages of the first packet evolution. The decay of the quasistationary state leads to the coming out of the secondary wave packets following behind the reflected and transmitted impulses.Reuse & Permissions

Figure 7

More complicated behavior and form of the scattering and of the secondary impulses accompanying the scattering of the second packet from Fig. 4b with spectral function covering the twenty-fourth line of the total transparency near the middle of the third transmission band [the inset in (b)].Reuse & Permissions

Figure 8

Evolution of the third wave packet from Fig. 4b.Reuse & Permissions

Figure 9

Evolution of the wave packet with the parameters of the fourth packet from Fig. 4b.Reuse & Permissions

Figure 10

Evolution of the spectrally wide fifth wave packet from Fig. 4b with spectral function covering two transmission bands.Reuse & Permissions

Figure 11

Points show the coordinates of the main peaks of the nonstationary probability density $\mid \Psi (x,t){\mid }^2$ plotted against time were evaluated from the numerical computations presented in Figs. 5, 6, 7, 8, 9, 10 [the numbers of initial packets by Fig. 4b are in ovals], as outside the lattice (1) falling, (2) reflected, and (5) transmitted or generated by the quasistationary states decaying secondary impulses coming out the lattice] so inside the lattice [(3) penetrating, (4) internally reflected or quasistationary] the subscripts enumerate the impulses of the same type following one another. The slopes of lines correspond to the group velocities and their shifts to the delay times. In (a) the lines $4_1$, $4_2$ connected with the existing and $2_2$, $5_2$ with the decaying of the quasistationary states are generated by the first packet, the lines 3 and $4_2$ intersect. Gray strip on (b) locates the region of the quasistationary states are generated by the second packet. Gray strip on (e) locates the tail region of the fifth packet. The variable $x$ is measured in angstroms Å and $t$ in atomic units.Reuse & Permissions

Figure 12

Poles $k_R$ (points in lower half-plane) and zeros $k_R^0$ (primes on the real axes) of the reflection amplitude $r=r\left(k\right)$ in the limits of the third transmission band were calculated for the parameters of lattice and the first wave packet from Fig. 4. The spectral function $C\left(k\right)$ and the integration contour with going around the poles that are closest to the saddle $k_S$ are also shown.Reuse & Permissions

Figure 13

Poles $k_R$ of the reflection amplitude $r=r\left(k\right)$ in the third transmission band were calculated for the same parameters of wave packet as in Fig. 12, but for the $\delta$ lattices with various $N=25$, 50, 75, and 100.Reuse & Permissions

Figure 14

Points show the probability $w_L$ that was calculated according to the dates of Figs. 5, 6. The exponential curve is the result of the two parameters fitting by one term from Eq. (6.2) with ${\tau }_{R2}=4450\text{a.u.}$, $t_{02}=550\text{a.u.}$, and with $w_{0\nu }=1$. The estimations by [Ref. 1, (4.8)] with coordinates of the second from the right pole on Fig. 12 give ${\tau }_{R2}=4494\text{a.e.}$; the first from the right pole gives a significantly larger value ${\tau }_{R1}=16000\text{a.u.}$, but its contribution in probability is small on that time interval because of a small spectral weight.Reuse & Permissions

Figure 15

Probability density after the scattering of the first wave packet from Fig. 4b and Fig. 11a at the moment $t=3.5\times {10}^4\text{a.u.}$ (a) The space oscillations from the beats of two resonance wave functions have the wavelength approximately equal to ${\lambda }_b\sim 2\pi ∕\mid k_{R01}-k_{R02}\mid \approx 2813\mathrm{\AA }$. The inset shows the probability density in enlarged scale. (b) Inside the lattice the wave function took a form of the first resonance state of the type that is shown in Fig. 3(6).Reuse & Permissions

Figure 16

Probability current for the same packet as in the previous figure: (a) At the point $x=0$ on the left end of the lattice; (b) at the point $x=Nd$ on the right end of the lattice. The inset show probability current in enlarged scale. The period of oscillations is approximately equal to $T_b\approx 2\pi ∕(E_{R1}-E_{R2})\approx 10680\text{a.e.}$Reuse & Permissions