Runge-Kutta methods), a very tiny time step must be chosen in order to properly account for the influence of the kneelike barrier; this, in fact, often leads to instability of the numerical scheme for a long time propagation. A physical reason of such an instability lies in the fact that nonsymplectic integrators do not explicitly conserve energy while the symplectic integrators always do; hence, they give a proper long-time evolution of any chaotic Hamiltonian system.

The observation of this behavior of the classical counterpart casts doubt on the quantum nature of the “paradox.” More precisely, is it possible that an ensemble of classical particles, which corresponds (in some sense) to the initial wave function of the system at hand (\[InitialCondition\]), would mimic the observed phenomenon? As shown below, the answer turns out to be negative.

Furthermore, since it is well known that the application of the semi classical approximation, as a mediator between classical and quantum mechanics, to tunneling often is very fruitful in shading light on the physical nature of the studied process (see, e.g., Refs. [@Maitra1997; @Spanner2003]), the addressed question is important as the first step toward the usage of semiclassical methods for the interpretation of the “paradox.”

