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A known limitation of time-dependent mean-field approaches is a lack ofquantum tunneling for collective motions such as in sub-barrier fusionreactions. As a first step toward a solution, a time-dependent model isconsidered using a superposition of Gaussian wave packets, to describe therelative motion between two colliding nuclei, which may be simplified to aproblem for one particle in one dimension. In this article, how the modeldescribes the potential-barrier transmission is investigated by payingattention to the time evolution of the phase space distribution, which inparticular reveals that the behavior of the free propagation of the incomingstate is not trivial, depending on the number of superposed wave packets.Passage over the barrier can occur due to the high-momentum components in theincoming state corresponding to energies above the barrier height, which is,however, of classical nature and needs to be distinguished from the truequantum tunneling. Although a transmitted wave packet in some case may end upwith an energy lower than the barrier, a difficulty is noticed in guaranteeingthe energy conservation when the energies of different exit channels, e.g. oftransmission and reflection, are individually measured. To overcome theseissues for a description of quantum tunneling is still a challenging problem.This article mainly treats the same system with the same model as in the paperPhys. Lett. B 808 (2020) 135693, arXiv:2006.06944v1 by N. Hasegawa, K. Haginoand Y. Tanimura. However, the conclusion of the present work disagrees withtheir quick conclusion that quantum tunneling was simulated by the model.Comments are made on this.

Phase-space consideration on barrier transmission in a time-dependent variational approach with superposed wave packets