Dividing Line between Quantum and Classical Trajectories: Bohmian Time Constant

TL;DR

This paper introduces a Bohmian time constant derived from a nonlinear Schrödinger equation, which marks the transition from quantum to classical behavior under continuous measurement and damping, with implications for understanding measurement in quantum mechanics.

Contribution

It proposes a generalized nonlinear Schrödinger equation within Bohmian mechanics to identify a time constant that delineates quantum and classical regimes.

Findings

The Bohmian time constant is approximately 10^{-26} seconds for an electron-sized wave packet.

Continuous measurement and damping cause quantum trajectories to exponentially converge to classical trajectories.

Damping suppresses quantum effects faster than the system's relaxation time.

Abstract

This work proposes an answer to a challenge posed by Bell on the lack of clarity in regards to the line between the quantum and classical regimes in a measurement problem. To this end, a generalized logarithmic nonlinear Schr\"odinger equation is proposed to describe the time evolution of a quantum dissipative system under continuous measurement. Within the Bohmian mechanics framework, a solution to this equation reveals a novel result: it displays a time constant which should represent the dividing line between the quantum and classical trajectories. It is shown that continuous measurements and damping not only disturb the particle but compel the system to converge in time to a Newtonian regime. While the width of the wave packet may reach a stationary regime, its quantum trajectories converge exponentially in time to classical trajectories. In particular, it is shown that damping tends to suppress further quantum effects on a time scale shorter than the relaxation time of the system. If the initial wave packet width is taken to be equal to 2.8 10^{-15} m (the approximate size of an electron), the Bohmian time constant is found to have an upper limit, i. e., ${\tau_{B\max}} = {10^{- 26}}s$.