Completeness of observables and particle trajectories in quantum mechanics

The complete set of observables (bilinear Hermitian forms) is determined for the Schrodinger equation and their connection with the curvature and torsion of the curves, where conservation laws are fulfilled, is established. It is shown that these curves for a free particle, in the general case, are spiral lines with the radius and step length defined by the observables at the initial point (both parameters are proportional to the de Broglie wavelength). A spiral line turns to a straight line under some conditions. The trajectory variations are considered in the problem with a potential step and a rectangular barrier. It is shown that spiral lines can be transformed into straight lines and vice versa. All observables, which are changed along the potential barrier, can be restored under some constraints on the potential. The Hermitian transformations at the potential step are connected with the Lorentz transformations. A qualitative explanation of the double-slit experiment for extremely low intensity of the particles' source in the absence of the interference conditions is suggested.