Optimal guidance law in quantum mechanics

• Treating quantum mechanics as a pursuit-evasion game.
• Reveal an interesting analogy between guided flight motion and guided quantum motion.
• Solve optimal quantum guidance problem by dynamic programming.
• Gives a formal proof of de Broglie–Bohm’s idea of a pilot wave.
• The optimal pilot wave is shown to be a wavefunction solved from Schrödinger equation.

Following de Broglie’s idea of a pilot wave, this paper treats quantum mechanics as a problem of stochastic optimal guidance law design. The guidance scenario considered in the quantum world is that an electron is the flight vehicle to be guided and its accompanying pilot wave is the guidance law to be designed so as to guide the electron to a random target driven by the Wiener process, while minimizing a cost-to-go function. After solving the stochastic optimal guidance problem by differential dynamic programming, we point out that the optimal pilot wave guiding the particle’s motion is just the wavefunction Ψ(t,x), a solution to the Schrödinger equation; meanwhile, the closed-loop guidance system forms a complex state–space dynamics for Ψ(t,x), from which quantum operators emerge naturally. Quantum trajectories under the action of the optimal guidance law are solved and their statistical distribution is shown to coincide with the prediction of the probability density function Ψ∗ΨΨ∗Ψ.