Wave packet dynamics for a non-linear Schrödinger equation describing continuous position measurements

• Time-dependent solutions for non-linear Schrödinger equation.
• Stability of stationary solutions for non-linear Schrödinger equation.
• Bohmian interpretation and position quantum measurement.

We investigate time-dependent solutions for a non-linear Schrödinger equation recently proposed by Nassar and Miret-Artés (NM) to describe the continuous measurement of the position of a quantum particle (Nassar, 2013; Nassar and Miret-Artés, 2013). Here we extend these previous studies in two different directions. On the one hand, we incorporate a potential energy term in the NM equation and explore the corresponding wave packet dynamics, while in the previous works the analysis was restricted to the free-particle case. On the other hand, we investigate time-dependent solutions while previous studies focused on a stationary one. We obtain exact wave packet solutions for linear and quadratic potentials, and approximate solutions for the Morse potential. The free-particle case is also revisited from a time-dependent point of view. Our analysis of time-dependent solutions allows us to determine the stability properties of the stationary solution considered in Nassar (2013), Nassar and Miret-Artés (2013). On the basis of these results we reconsider the Bohmian approach to the NM equation, taking into account the fact that the evolution equation for the probability density ρ=|ψ|2ρ=|ψ|2 is not a continuity equation. We show that the effect of the source term appearing in the evolution equation for ρρ has to be explicitly taken into account when interpreting the NM equation from a Bohmian point of view.