Classical trajectories compatible with quantum mechanics

Abstract

Consider any stationary Schroedinger wave equation (SWE) solution $psi(x)$ for a particle. The corresponding PDF on position QTR{em}{x} of the particle is QTR{em}{p}${}_X(x)=|psi(x){|}^2$. 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}$pi)sin(2pix/L)+t_0=t.$ The constant QTR{em}{t}${}_0$ 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.

References in corpus (1)

• Bohmian Mechanics Revisited