A deterministic state-space model of intrinsic quantum uncertainties

For an intrinsic quantum uncertainty described by |x - x̅(t)| ≤ Δx and |p - p̅(t)|≤ Δp, with ΔxΔp ≥ h/2, we establish a state-space model governed by the quantum Hamilton equations q̇ = f(q, p) and ṗ == g(q, p) such that their solutions (q(t), p(t)) satisfy the given uncertainty bound and meanwhile, the statistical distributions of the trajectory sets {q(t)} and {p(t)} exactly reproduce the probability density functions ψ* (x)ψ(x) and ψ* (p)ψ(p) prescribed a priori. Although actual quantum trajectories are random and non-differentiable, we point out that a complex trajectory q(t) solved from the state-space model provides an uncertainty boundary encompassing an ensemble of quantum trajectories in such a way that its real part q_R (t) provides the mean trajectory, while its imaginary part q₁ (t) gives the radius of deviation of the quantum trajectories from the mean trajectory.