^{1}and Brad A. Rowland

^{1}

### Abstract

The quantum Hamilton-Jacobi equation for the action function is approximately solved by propagating individual Lagrangian quantum trajectories in complex-valued phase space. Equations of motion for these trajectories are derived through use of the derivative propagation method (DPM), which leads to a hierarchy of coupled differential equations for the action function and its spatial derivatives along each trajectory. In this study, complex-valued classical trajectories (second order DPM), along which is transported quantum phase information, are used to study low energy barrier transmission for a model two-dimensional system involving either an Eckart or Gaussian barrier along the reaction coordinate coupled to a harmonic oscillator. The arrival time for trajectories to reach the transmitted (product) region is studied. Trajectories launched from an “equal arrival time surface,” defined as an isochrone, all reach the real-valued subspace in the transmitted region at the same time. The Rutherford-type diffraction of trajectories around poles in the complex extended Eckart potential energy surface is described. For thin barriers, these poles are close to the real axis and present problems for computing the transmitted density. In contrast, for the Gaussian barrier or the thick Eckart barrier where the poles are further from the real axis, smooth transmitted densities are obtained. Results obtained using higher-order quantum trajectories (third order DPM) are described for both thick and thin barriers, and some issues that arise for thin barriers are examined.

The authors thank Chia-Chun Chou and Julianne Kuck David for helpful comments. This work was supported in part by the Robert Welch Foundation of Houston, Texas.

I. INTRODUCTION

II. MODEL TWO-DIMENSIONAL SCATTERING PROBLEM

A. Potential energy surfaces in complex coordinate space

B. Wave packet in complex coordinate space

III. SOLVING THE QUANTUM HAMILTON-JACOBI EQUATION

A. Lagrangianequation for the action

B. Derivative propagation along quantum trajectories

IV. COMPUTATIONAL RESULTS

A. Distribution of arrival times for transmitted trajectories

B. Arrival times and isochrones

C. Model for isochrones

D. Diffraction of trajectories around poles

E. Transmitted wave packet for thin barriers

F. Transmitted wave packet for the thick barrier

G. Results for quantum trajectories

H. Computer time requirements

V. SUMMARY

### Key Topics

- Lagrangian mechanics
- 26.0
- Potential energy surfaces
- 21.0
- Equations of motion
- 12.0
- Hamilton-Jacobi equations
- 10.0
- Wave functions
- 10.0

## Figures

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.

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.

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 , , , .

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 , , , .

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 .

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 .

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

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

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.

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.

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

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

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.

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.

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.

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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