In recent years,intensive effort has gone into developing numerical tools for exact quantum mechanical calculations that are based on Bohmian mechanics.As part of this effort we have recently developed as alternative formulation of Bohmian mechanics in which the quantum action 5 is taken to be complex [Y.Goldfarb et al,J.Chem.Phys.125,231103 (2006)].In the alternative formulation there is a significant reduction in the magnitude of the quantum force as compared with the conventional Bohmian formulation,at the price of propagating complex trajectories.In this paper we show that Bohmian mechanics with complex action is able to overcome the main computational limitation of conventional Bohmian methods-the propagation of wave functions once nodes set in.In the vicinity of nodes,the quantum force in conventional Bohmian formulations exhibits rapid oscillations that present a severe numerical challenge.We show that within complex Bohmian mechanics,multiple complex initial conditions can lead to the same real final position,allowing for the accurate description of nodes as a sum of the contribution from two or more crossing trajectories.The idea is illustrated on the reflection amplitude from a one-dimensional Eckart barrier.We believe that trajectory crossing,although in contradiction to the conventional Bohmian trajectory interpretation,provides an important new tool for dealing with the nodal problem in Bohmian methods.
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机译：近年来，人们已经投入大量精力来开发基于Bohmian力学的精确量子力学计算的数值工具。作为这一努力的一部分，我们最近开发了Bohmian力学的替代公式，其中量子作用5被认为是复杂的[Y.Goldfarb et al。，J.Chem.Phys.125,231103（2006）]。在替代配方中，与传统的波门公式相比，量子力的大小显着降低，但传播的代价本文证明了具有复杂作用的Bohm力学能够克服传统Bohmian方法的主要计算限制-节点一旦进入波函数的传播。在节点附近，常规Bohmian公式中的量子力表现出快速的振荡，这对数值提出了严峻的挑战。我们表明，在复杂的波姆力学中，多个复杂的初始条件可能导致到相同的实际最终位置，从而可以精确地将节点描述为两个或多个交叉轨迹的贡献之和。该思想在一维Eckart势垒的反射幅度上得到了说明。尽管如此，我们相信轨迹交叉与传统的波门轨迹解释相反，它为处理波门方法中的节点问题提供了重要的新工具。
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