Quantum trajectories and the Bohm time constant

This work proposes a new logarithmic nonlinear Schrödinger equation to describe the dynamics of a wave packet under continuous measurement. Via the method of quantum trajectories formalism of the Bohmian model of quantum mechanics, it is shown that this continuous measurement alters the dynamical properties of the measured system. While the width of the wave packet may reach a stationary regime, its quantum trajectories converge asymptotically in time to classical trajectories. So, continuous measurements not only disturb the particle but compel it to eventually converge to a Newtonian regime. The rate of convergence depends on what is defined here as the Bohm time constant which characterizes the resolution time of the measurement. If the initial wave packet width is taken to be equal to 2.8×10−15 m (the approximate size of an electron) then the Bohm time constant is found to be about 6.8×10−26 s.

► New nonlinear Schroedinger equation is proposed to describe continuous measurement.
► Equation is solved via the Bohmian model of quantum mechanics.
► Quantum trajectories of wave packet converge in time to classical trajectories.
► Bohm time constant (characteristic time of the measurement) is calculated.