Classical trajectories compatible with quantum mechanics
Abstract
Consider any stationary Schroedinger wave equation (SWE) solution for a particle. The corresponding PDF on position QTR{em}{x} of the particle is QTR{em}{p}. There is a classical trajectory QTR{em}{x(t)} for the particle that is consistent with this PDF. The trajectory is unique to within an additive constant corresponding to an initial condition QTR{em}{x(0).} However the value of QTR{em}{x(0)} cannot be known. As an example, a free particle in its ground state in a box of length QTR{em}{L} obeys a classical trajectory QTR{em}{x/L - (1/2} The constant QTR{em}{t} is an unknowable time displacement. Momentum values, however, cannot be determined by merely differentiating QTR{em}{d/dt} the trajectory QTR{em}{x(t)} and, instead, follow the usual quantification rules of Heisenberg's. This permits position and momentum to remain complementary variables. Our approach is fundamentally different from that of D. Bohm.
Cite
@article{arxiv.quant-ph/0006012,
title = {Classical trajectories compatible with quantum mechanics},
author = {B. Roy Frieden and A. Plastino},
journal= {arXiv preprint arXiv:quant-ph/0006012},
year = {2009}
}