Computing the wavefunction from trajectories: particle and wave pictures in quantum mechanics and their relation

Abstract We describe the particle method in quantum mechanics which provides an exact scheme to calculate the time-dependent wavefunction from a single-valued continuum of determinstic trajectories where two spacetime points are linked by at most a single orbit. A natural language for the theory is offered by the hydrodynamic analogy, in which wave mechanics corresponds to the Eulerian picture and the particle theory to the Lagrangian picture. The Lagrangian model for the quantum fluid may be developed from a variational principle. The Euler–Lagrange equations imply a fourth-order nonlinear partial differential equation to calculate the trajectories of the fluid particles as functions of their initial coordinates using as input the initial wavefunction. The admissible solutions are those consistent with quasi-potential flow. The effect of the superposition principle is represented via a nonclassical force on each particle. The wavefunction is computed via the standard map between the Lagrangian coordinates and the Eulerian fields, which provides the analogue in this model of Huygens’ principle in wave mechanics. The method is illustrated by calculating the time-dependence of a free Gaussian wavefunction. The Eulerian and Lagrangian pictures are complementary descriptions of a quantum process in that they have associated Hamiltonian formulations that are connected by a canonical transformation. The de Broglie–Bohm interpretation, which employs the same set of trajectories, should not be conflated with the Lagrangian version of the hydrodynamic interpretation. The theory implies that the mathematical results of the de Broglie–Bohm model may be regarded as statements about quantum mechanics itself rather than about its interpretation.