Alternative classical trajectories compatible with quantum mechanics

Abstract

Consider any solution ψ( x| t) to the Schrödinger wave equation (SWE) for a single, non-relativistic particle. The corresponding PDF on position x of the particle is p X ( x| t)=| ψ( x| t)| 2. We show that one may construct a classical trajectory x( t) for the particle whose histogram of values x over time precisely follows this PDF. The trajectory generally differs from the celebrated Bohm trajectory under the same conditions. As an example of the approach, a free particle in a box in a definite energy state E has a ground-state solution ψ(x|t)= 2/L cos(πx/L) exp(iEt/ℏ) , | x|⩽ L/2. Its corresponding PDF is p X ( x| t)=(2/ L)cos 2( πx/ L). The particle's trajectory x( t) then obeys x/ L+(2 π) −1sin(2 πx/ L)=( t− t 0)/ T at each t. Each trajectory is unique to within an additive constant corresponding to an initial condition x(0) or t 0 that cannot be known. Similarly, a classical “trajectory” μ( t) in the space of momentum values μ can be found that agrees with any required SWE momentum eigenfunction for the particle. The approach is completely generalized to a scenario of multiple particles obeying a relativistic wave equation.