A Random Map Description for Quantum Superposition

Chaotic maps are deterministic yet asymptotically in time behave in a statistical manner. In this note we present a chaotic dynamical model that is consistent with observable quantum mechanics and succeeds in presenting a physical process for superposition of wave functions, namely chaotic random maps. In place of the wavefuction we shall use real chaotic maps as the underlying mechanism for the observed probability density functions. Let ψi(x, t), i = 1, 2 be two eigenfunctions of a quantum mechanical par-ticle system. We associate with each ψi(x, t) a deterministic nonlinear point transformation τ i(x) whose unique invariant probability density function is the observed density ρi(x, t) = ψ i (x, t)ψi(x, t). We consider the superposed wavefunction ψ(x, t) = aψ1(x, t) + bψ2(x, t) and show that we can associate with ψ(x, t), a random chaoitic map related to τ1(x)and τ2(x), whose in-variant probability density function ft(x) is equal to ψ ∗(x, t)ψ(x, t), where t denotes time. This description allows for a physical interpretation of quan-tum superposition. Numerical simulations of a two-slit experiment is done ∗The research has been supported by NSERC grants. 1 which show that the random map dynamics achieves the interference super-position pattern very accurately