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Quantum trajectories in complex phase space: Multidimensional barrier transmission
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10.1063/1.2746869
/content/aip/journal/jcp/127/4/10.1063/1.2746869
http://aip.metastore.ingenta.com/content/aip/journal/jcp/127/4/10.1063/1.2746869
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

Isosurfaces for the real part of the thin (a) Eckart and (b) Gaussian potential energy surfaces plotted in the three-dimensional subspace, with . The two isosurfaces in each figure have the values The vertical tubes in (a) enclose poles in the complex-valued Eckart potential energy surface. The Gaussian potential surface in (b) exhibits oscillations along the direction, but there are no poles.

Image of FIG. 2.
FIG. 2.

Isosurfaces of the absolute value of the complex-valued wave function plotted in the three-dimensional subspace, with . The four isosurfaces have the values , , , and . The center of the wave packet is given by , , , .

Image of FIG. 3.
FIG. 3.

Distribution of arrival times for transmitted trajectories when . The arrival times are plotted for starting points in the translational subspace. These times were calculated for trajectories evolving on the thin Eckart potential energy surface.

Image of FIG. 4.
FIG. 4.

Arrival times for trajectories launched with . The gap between early and late arrival time trajectories is evident between and . These arrival times were computed for trajectories evolving on the thin Eckart potential surface.

Image of FIG. 5.
FIG. 5.

Three isochrones for wave packet scattering on the thin Eckart potential surface at . For three arrival times, , 1200, and 1800, vertical lines down to the plane locate the initial translational coordinates for the trajectories (the isochrones). Trajectories launched from an isochrone reach the real-valued subspace of the transmitted region at the same time.

Image of FIG. 6.
FIG. 6.

Modulus of the arrival time, , for 17 values of the initial phase angle (measured counterclockwise from the positive axis). For these trajectories, the initial amplitude of the oscillator was and . These trajectories were run on the thin Eckart barrier potential energy surface.

Image of FIG. 7.
FIG. 7.

Isochrones for two initial vibrational amplitudes, and 0.5. The trajectory results are shown by dots and cubic polynomial fits are shown as curves. These trajectories were run on the Gaussian potential energy surface.

Image of FIG. 8.
FIG. 8.

Two trajectories, computed on the thin Eckart barrier potential surface, diverting around the pole in the potential energy at . Three isosurfaces of the absolute value of the potential are also shown. The trajectory launched from diverts to the right around the pole while the second trajectory, launched from , diverts to the left. Both trajectories start with the vibrational coordinates and and they have the same arrival time, . Projections of the trajectories in the lower horizontal plane are also shown.

Image of FIG. 9.
FIG. 9.

Projection into the plane of early and late arrival time trajectories launched from . The early arrival trajectories , launched from starting positions in the interval to , divert up around the pole. The late arrival time trajectories , launched from starting positions in the interval to , divert downward around the pole. Trajectories launched within the gap from to do not cross the real plane on the product side of the barrier. Contours of the absolute value of the thin Eckart potential are also shown.

Image of FIG. 10.
FIG. 10.

Probability densities (dots) at the trajectory positions for the transmitted wave packet when the energy is . These trajectories propagate on the thin Eckart potential surface for the arrival time . (a) View from the back side of the transmitted packet. (b) Side view of the packet, showing the sudden falloff in density near .

Image of FIG. 11.
FIG. 11.

Probability densities (dots) at the trajectory positions for the transmitted wave packet. These trajectories propagate on the Gaussian potential surface for the arrival time .

Image of FIG. 12.
FIG. 12.

Absolute value of the initial wave function and the thin and thick Eckart barriers plotted along the real axis. The wave function has been scaled to have the same height as the potential barrier.

Image of FIG. 13.
FIG. 13.

Quantum trajectories computed on the thin Gaussian potential energy surface: CVDPM(2) and CVDPM(3). These trajectories were launched with the initial vibrational amplitude and they pass through the real product plane at the arrival time . The propagation time was

Image of FIG. 14.
FIG. 14.

Probability densities computed for trajectories on the thick Eckart potential energy surface: CVDPM(2) results (large dots) and CVDPM(3) results (small dots). Exact results from a large fixed grid calculation are also shown (curve). These densities are shown along the slice through the transmitted wave packet.

Image of FIG. 15.
FIG. 15.

Probability densities computed for trajectories on the intermediate width Gaussian potential energy surface: polynomial fit to CVDPM(2) results (large dots) and fit to CVDPM(3) results (small dots). Exact results from a large fixed grid calculation are also shown (curve). These densities are shown along the slice through the transmitted wave packet.

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/content/aip/journal/jcp/127/4/10.1063/1.2746869
2007-07-24
2014-04-16
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Quantum trajectories in complex phase space: Multidimensional barrier transmission
http://aip.metastore.ingenta.com/content/aip/journal/jcp/127/4/10.1063/1.2746869
10.1063/1.2746869
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