English

Classical trajectories compatible with quantum mechanics

Quantum Physics 2009-11-06 v1

Abstract

Consider any stationary Schroedinger wave equation (SWE) solution psi(x)psi (x) for a particle. The corresponding PDF on position QTR{em}{x} of the particle is QTR{em}{p}X(x)=psi(x)2_{X}(x)=|psi (x)|^{2}. 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}pi)sin(2pix/L)+t0=t.pi)sin (2pi x/L)+t_{0}=t. The constant QTR{em}{t}0_{0} 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.

Keywords

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}
}