ABSTRACT
Transcending its origin in the debate on the interpretation of quantum theory, the continuous deterministic trajectory has proved a potent tool in computational quantum mechanics (an excellent introduction to numerical trajectory techniques is provided by Wyatt’s text [1]). It is straightforward to develop a method to derive the timedependent Schrödinger equation from a single-valued continuum of spacetime trajectories, and supply a corresponding exact formula for the wavefunction [2].Anatural language for this theory of evolution is offered by the hydrodynamic analogy, in which wave mechanics corresponds to the Eulerian picture (as shown by Madelung [3]) and the trajectory theory to the Lagrangian picture. The Lagrangian model for the quantum fluid may be developed from a variational principle involving a specific interaction potential (the quantum internal potential energy), and the Euler-Lagrange equations imply a nonlinear partial differential equation to calculate the trajectories of the fluid particles as functions of their initial coordinates and the initial wavefunction. The latter supplies two sets of data to the particle model: the initial density (via the amplitude) and the initial velocity (via the phase). The time-dependent wavefunction is then computed via the standard map between the Lagrangian coordinates and the Eulerian fields.
