  We follow up the work, where in light of the Picard-Lefschetz thimble
approach, we split up the real-time path integral into two parts: the initial
density matrix part which can be represented via an ensemble of initial
conditions, and the dynamic part of the path integral which corresponds to the
integration over field variables at all later times. This turns the path
integral into a two-stage problem where, for each initial condition, there
exits one and only one critical point and hence a single thimble in the complex
space, whose existence and uniqueness are guaranteed by the characteristics of
the initial value problem. In this paper, we test the method for a fully
quantum mechanical phenomenon, quantum tunnelling in quantum mechanics. We
compare the method to solving the Schr\"odinger equation numerically, and to
the classical-statistical approximation, which emerges naturally in a
well-defined limit. We find that the Picard-Lefschetz result matches the
expectation from quantum mechanics and that, for this application, the
classical-statistical approximation does not.
