Complex-valued derivative propagation method with approximate Bohmian trajectories for quantum barrier scattering

- The complex quantum Hamilton-Jacobi equation is approximately solved in real space.
- Equations of motion are derived through use of the derivative propagation method.
- Numerically unstable reflected trajectories may pass through the potential barrier.
- Transmitted wave packet is obtained by propagating individual Bohmian trajectories.
- Excellent transmission probabilities are obtained for both thick and thin barriers.

The complex quantum Hamilton-Jacobi equation for the complex action is approximately solved by propagating individual Bohmian trajectories in real space. Equations of motion for the complex action and its spatial derivatives are derived through use of the derivative propagation method. We transform these equations into the arbitrary Lagrangian-Eulerian version with the grid velocity matching the flow velocity of the probability fluid. Setting higher-order derivatives equal to zero, we obtain a truncated system of equations of motion describing the rate of change in the complex action and its spatial derivatives transported along approximate Bohmian trajectories. A set of test trajectories is propagated to determine appropriate initial positions for transmitted trajectories. Computational results for transmitted wave packets and transmission probabilities are presented and analyzed for a one-dimensional Eckart barrier and a two-dimensional system involving either a thick or thin Eckart barrier along the reaction coordinate coupled to a harmonic oscillator.

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