Non-dynamical quantum trajectories

Commonly held opinion is that particle trajectory descriptions are incompatible with quantum mechanics. Louis de Broglie (1926) first proposed a way to include trajectories in quantum mechanics, but the idea was abandoned until David Bohm (1952) re-invented and improved the theory. Bohm interprets the particle trajectories as physically real; for example, an electron actually is a particle moving on a well defined trajectory with a position and momentum at all times. By design, Bohm's trajectories never make predictions that differ from standard quantum mechanics, and their existence cannot be experimentally verified.

Three new methods to obtain Bohm's particle trajectories are presented. The methods are non-dynamical, and utilize none of Bohm's equations of motion; in fact, two of the methods have no equations for a particle's trajectory. Instead, all three methods use only the evolving probability density ρ=ψ*ψ to extract the trajectories. The first two methods rest upon probability conservation and density sampling, while the third method employs the informational or geometrical construction of centroidal Voronoi tessellations. In one-dimension all three methods are proved to be equivalent to Bohm's particle trajectories. For higher dimensional configuration spaces, the first two methods can be used in limited situations, but the last method can be applied in all cases. Typically, the resulting higher dimensional non-dynamical trajectories are also identical to Bohm.

Together the three methods point to a new interpretation of Bohm's particle trajectories, namely, the Bohm trajectories are simply a kinematic portrayal of the evolution of the probability density. In addition, the new methods can be used to measure Schrödinger's wave function and Planck's constant.