\[Fig\_potentials\](b); (b) comparison of Fig. \[Fig\_potentials\](c) vs. Fig. \[Fig\_potentials\](d); (c) comparison of Fig. \[Fig\_potentials\](e) vs. Fig. \[Fig\_potentials\](f).[]{data-label="Fig_probability_tunnel_time"}](ProbTunnelingTime.pdf)

Explanation of the effect {#Sec3}
=========================

The “paradox” posed in Sec. \[Sec2\] is resolved in this section by analyzing a perturbation theory solution of the Schrödinger equation.

In this section, we shall study a general two-1D-particles system given by the Hamiltonian $$\begin{aligned}
\label{HamiltonianR_RHO}
\hat{H} &=& \frac{-1}{2M}\frac{\partial^2}{\partial R^2} + \frac{-1}{2\mu}\frac{\partial^2}{\partial \rho^2} + \nonumber\\
&& U(\rho) + V_1\left( R - \frac{\mu}{m_1} \rho\right) + V_2\left( R + \frac{\mu}{m_2} \rho\right),\end{aligned}$$ which is already written in the (general) center of mass ($R$) and relative ($\rho$) coordinates, $$\begin{aligned}
&& R = (m_1 x_1 + m_2 x_2)/(m_1 + m_2), \quad \rho = x_2 -x_1, \nonumber\\
&& \mu = m_1 m_2 /M, \quad M = m_1 + m_2, \nonumber\end{aligned}$$ where $x_{1,2}$ ($m_{1,2}$), as previously, being the coordinates (masses) of the first and second particles, respectively.

Let us introduce the following notation $$\begin{aligned}
\hat{U}(t_f, t_i) &=& \hat{T} \exp\left[ -i \hat{H}(t_f - t_i)\right], \\
\hat{U}_0(t_f, t_i) &=& \hat{T} \exp\left[ -i( \hat{H} - V_1 - V_2)(t_f - t_i) \right],\end{aligned}$$ for total and unperturbed propagators, respectively. The sum of the potential barriers, $V_1 + V_2$, shall be considered as a perturbation. The eigenfunctions, ${\left| n 