Quantum localization due to mirror plane symmetry

With suitably directed initial states, we show how the time-averaged density of an evolving wave packet localizes on a mirror plane. A classical analog of this behavior can sometimes be found with trajectories weighted according to a Wigner distribution of the initial quantum state. However, in the limit of strongly chaotic classical dynamics, no such classical analog exists and the quantum localization in the density tends to be stronger by a factor of 2. Two very different systems are used to illustrate this effect, one being a three-dimensional model for a lithium atom moving within a ${\text{C}}_{60}$ cage and the other being a two-dimensional double well problem.

Figure 1

(Color online) (a) Image map of the potential at $r=1.54\text{Å}$. The energy scale is in ${\text{cm}}^{-1}$ relative to the minimum energy. (b) The density defined in Eq. (1) of a wave packet with mean energy $2000{\text{cm}}^{-1}$ after ${10}^4\text{a}\text{.u}\text{.}$ of time evolution. The circle is the position of the initial Gaussian wave packet and the arrow shows the direction its initial momentum vector. The angles $\theta$ and $\varphi$ in this and all subsequent figures are expressed in radians.

Figure 2

(Color online) The time-averaged polar density, $N\left(\theta ,\varphi \right)$, is imaged as a polar plot in (a) and on the two sphere in (b). The high density curve in (a) or (b) can be viewed as a great circle with Li moving over potential wells corresponding to the indicated pattern of ${\text{C}}_6$ and ${\text{C}}_5$ rings in (c). $\left(R=R^{\prime }\right)$. The initial momentum is chosen such that, when projected on to the cage, the wave packet would start around P, the center of one ${\text{C}}_6$ ring, and head roughly toward Q, which corresponds to the center of an adjacent ${\text{C}}_6$ well.

Figure 3

(Color online) Classical time-averaged polar densities, $N\left(\theta ,\varphi \right)$, for (a) the slightly misaligned case and (b) the perfectly aligned case.

Figure 4

(Color online) Comparison of the quantum (red curve) and classical (light green curve) integrated densities, $I_c\left(t\right)$, corresponding to the example of Fig. 3. The slightly misaligned initial wave packet case is in (a) and the perfectly aligned initial wave packet case is in (b). In the long time limit the quantum result is approximately twice the classical one in both cases.

Figure 5

(Color online) (a) The potential of Eq. (3). (b) The initial Gaussian wave packet. The time-averaged (c) quantum and (d) classical densities for motion of a unit mass particle with the given initial condition. The classical results are inferred from integration of trajectories weighted according to the Wigner distribution function of the initial Gaussian wave packet. (The units in this figure and Fig. 6 can be considered to be atomic units with $\hslash =1$.)

Figure 6

(Color online) Cuts through the quantum (red curve) and classical (light green curve) time-averaged densities of Figs. 5c, 5d corresponding to $y=0$. On the mirror plane $x=0$, the quantum result is approximately twice the classical one.