Abstract The ensemble mean equations for a classical particle moving stochastically obtain the form of fluid equations. When applying the Madelung transformation to write the Schrodinger equation in a fluid-like form we find that the equations are equivalent to the classical ensemble mean equations if an additional force is added to the equations. The latter can be expressed as a pressure gradient force of a fluctuating pressure with zero mean. Here we analyze the mechanism of quantum tunneling through a rectangular potential barrier from this perspective. We find that despite of the vanishing of the mean of the pressure fluctuations their local non zero gradients enable the tunneling by balancing the counter external potential gradients at the two sides of the potential barrier. Consequently, for stationary solutions, the ensemble mean kinetic energy remains unchanged across the boundaries of the barrier.

中文翻译：

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量子隧穿的有效经典随机理论

摘要 经典粒子随机运动的系综平均方程得到流体方程的形式。当应用马德隆变换以类似流体的形式写出薛定谔方程时，我们发现如果在方程中添加额外的力，这些方程等效于经典的系综平均方程。后者可以表示为均值为零的波动压力的压力梯度力。在这里，我们从这个角度分析了通过矩形势垒的量子隧穿机制。我们发现，尽管压力波动的平均值消失了，但它们的局部非零梯度通过平衡势垒两侧的反向外部电位梯度来实现隧穿。因此，对于平稳解，