Abstract:
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 timedependent
solutions while previous studies focused on a stationary
one.Weobtain 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 theNMequation,
taking into account the fact that the evolution equation for the
probability density ρ = |ψ|2 is not a continuity equation.Weshow
that the effect of the source term appearing in the evolution equation
for ρ has to be explicitly taken into account when interpreting
the NM equation from a Bohmian point of view.
