On quantum potential dynamics

Non-relativistic de Broglie-Bohm theory describes particles moving under the guidance of the wave function. In deBroglie’s original formulation, the particle dynamics is given by a first-order differential equation. In Bohm’s reformulation, it is given by Newton’s law of motion with an extra potential that depends on the wave function—the quantum potential—together with a constraint on the possible velocities. It was recently argued, mainly by numerical simulations, that relaxing this velocity constraint leads to a physically untenable theory. We provide further evidence for this by showing that for various wave functions the particles tend to escape the wave packet. In particular, we show that for a central classical potential and bound energy eigenstates the particle motion is often unbounded. This work seems particularly relevant for ways of simulating wave function evolution based on Bohm’s formulation of the de Broglie-Bohm theory. Namely, the simulations may become unstable due to deviations from the velocity constraint. Non-relativistic de Broglie-Bohm theory (also called Bohmian mechanics) [1–3] describes point-particles moving under the guidance of the wave function. In the case of spinless particles, with positions Xk, k = 1, . . . , n, and configuration X = (X1, . . . ,XN), the equations of motion are given by dXk(t) dt = 1 mk ∇kS(X(t), t) , (1) where the wave function ψ = |ψ(x, t)|eiS(x,t)/~, with x = (x1, . . . ,xn), satisfies the nonrelativistic Schrödinger equation