Exploring the nonclassical dynamics of the “classical” Schrödinger equation

The introduction of nonlinearities into the Schrödinger equation has been considered in the literature as an effective manner to describe the action of external environments or mean fields. Here, we explore the nonlinear effects induced by subtracting a term proportional to Bohm’s quantum potential to the usual (linear) Schrödinger equation, which generates the so-called “classical” Schrödinger equation. Although a simple nonlinear transformation allows us to recover the well-known classical Hamilton–Jacobi equation, by combining a series of analytical results (in the limiting cases) and simulations (whenever the analytical treatment is unaffordable), we find an analytical explanation to why the dynamics in the nonlinear “classical” regime is still strongly nonclassical. This is even more evident by establishing a one-to-one comparison between the Bohmian trajectories associated with the corresponding wave function and the classical trajectories that one should obtain. Based on these observations, it is clear that the transition to a fully classical regime requires extra conditions in order to remove any trace of coherence, which is the truly distinctive trait of quantum mechanics. This behavior is investigated in three paradigmatic cases, namely, the dispersion of a free propagating localized particle, the harmonic oscillator, and a simplified version of Young’s two-slit experiment. [Display omitted] •Exact analytical solutions to the classical Schrödinger equation are provided.•The transition from linear/quantum regimes to nonlinear/classicals ones is analyzed.•Numerical simulations show evidence of nonclassical behaviors in classical regimes.•The (non)classical limit of Bohmian mechanics is shown and discussed.•Self-focusing induced by nonlinearities in harmonic potentials is demonstrated.