WKB analysis of Bohmian dynamics

Abstract

We consider a semi-classically scaled Schrödinger equation with WKB initial data. We prove that in the classical limit the corresponding Bohmian trajectories converge (locally in measure) to the classical trajectories before the appearance of the first caustic. In a second step we show that after caustic onset this convergence in general no longer holds. In addition, we provide numerical simulations of the Bohmian trajectories in the semiclassical regime which illustrate the above results.

Quantum Raychaudhuri equation

“…In the above, ψ itself evolves according to the Schrödinger equation. Analytical as well as numerical studies indeed demonstrate such interaction between wave packets, between a wave packet and a barrier etc, at short distances, and that although they can come close to each other, they never actually meet or cross [3,[7][8][9]. One may think of this as an effective repulsion between trajectories at short distances, due to the quantum potential.…”

“…Furthermore, although Eqs. (7) and (8) are completely equivalent to the Schrödinger equation, they can be interpreted as giving rise to actual trajectories of particles ("quantal trajectories") initially distributed according to the density |ψ| 2 , and in quantum equilibrium, subject to the external potential V ( x, t), as well as the additional quantum potential V Q . Indeed the latter reproduces the observed interference patterns in a double slit experiment, the Aharonov-Bohm effect, the Stern-Gerlach-type experiments, and all other observed quantum phenomena, and so long as quantum mechanics is valid, no experiments or observations can invalidate the above picture [3].…”

“…( 6) is first order in time, and at any time t, its right-hand side, and therefore the velocity field are uniquely defined at each point in space. Therefore it gives rise to nonintersecting integral curves or streamlines [3,7,8]. In the above, ψ itself evolves according to the Schrödinger equation.…”

Reply to “Comment on ‘Quantum Raychaudhuri equation’”

“…Next, quantal trajectories have the following wellknown properties: (a) Two such trajectories do not meet or cross [3][4][5][6][7][8][9]. Hence they do not form conjugate points in a given manifold.…”