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See detailBohmian trajectories for the half-line barrier
Dubertrand, Rémy ULiege; Shim, J.-B.; Struyve, W.

in Journal of Physics. A, Mathematical and Theoretical (2018), 51(8), 085302

Bohmian trajectories are considered for a particle that is free (i.e. the potential energy is zero), except for a half-line barrier. On the barrier, both Dirichlet and Neumann boundary conditions are ... [more ▼]

Bohmian trajectories are considered for a particle that is free (i.e. the potential energy is zero), except for a half-line barrier. On the barrier, both Dirichlet and Neumann boundary conditions are considered. The half-line barrier yields one of the simplest cases of diffraction. Using the exact time-dependent propagator found by Schulman, the trajectories are computed numerically for different initial Gaussian wave packets. In particular, it is found that different boundary conditions may lead to qualitatively different sets of trajectories. In the Dirichlet case, the particles tend to be more strongly repelled. The case of an incoming plane wave is also considered. The corresponding Bohmian trajectories are compared with the trajectories of an oil drop hopping on the surface of a vibrating bath. © 2018 IOP Publishing Ltd. [less ▲]

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See detailThe quantum separability problem is a simultaneous hollowisation matrix analysis problem
Neven, Antoine ULiege; Bastin, Thierry ULiege

in Journal of Physics. A, Mathematical and Theoretical (2018), 51

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See detailChaotic Bohmian trajectories for stationary states
Cesa, Alexandre ULiege; Martin, John ULiege; Struyve, Ward

in Journal of Physics. A, Mathematical and Theoretical (2016), 49

In Bohmian mechanics, the nodes of the wave function play an important role in the generation of chaos. However, so far, most of the attention has been on moving nodes; little is known about the ... [more ▼]

In Bohmian mechanics, the nodes of the wave function play an important role in the generation of chaos. However, so far, most of the attention has been on moving nodes; little is known about the possibility of chaos in the case of stationary nodes. We address this question by considering stationary states, which provide the simplest examples of wave functions with stationary nodes. We provide examples of stationary wave functions for which there is chaos, as demonstrated by numerical computations, for one particle moving in 3 spatial dimensions and for two and three entangled particles in two dimensions. Our conclusion is that the motion of the nodes is not necessary for the generation of chaos. What is important is the overall complexity of the wave function. That is, if the wave function, or rather its phase, has complex spatial variations, it will lead to complex Bohmian trajectories and hence to chaos. Another aspect of our work concerns the average Lyapunov exponent, which quantifies the overall amount of chaos. Since it is very hard to evaluate the average Lyapunov exponent analytically, which is often computed numerically, it is useful to have simple quantities that agree well with the average Lyapunov exponent. We investigate possible correlations with quantities such as the participation ratio and different measures of entanglement, for different systems and different families of stationary wave functions. We find that these quantities often tend to correlate to the amount of chaos. However, the correlation is not perfect, because, in particular, these measures do not depend on the form of the basis states used to expand the wave function, while the amount of chaos does. [less ▲]

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