Following the de Broglie's idea of 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 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 Schrödinger equation. 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*ψ.
Following the de Broglie's idea of 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 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 Schrödinger equation. 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*ψ.
In an article written by Louis de Broglie in 1959 (30 years after the Nobel prize rewarding his foundation of Wave Mechanics), the most challenging problem raised by the Bohr, Heisenberg and Born Standard Quantum Mechanics (SQM) was pointed out in the renunciation to describe “a permanent localization in space, and therefore a well-defined trajectory” for any moving particle. This challenge is taken up in the present paper, showing that de Broglie’s Primary Assumption p=hk, predicting the wave-particle duality, does also allow to obtain from the energy-dependent form of the Schrödinger and/or Klein-Gordon equations the Guidance Laws piloting particles along well-defined trajectories. The energy-independent equations, on the other hand, may only give rise—both in SQM and in the Bohmian approach—to probabilistic descriptions, overshadowing the role of de Broglie’s matter waves in physical space.
In a pilot-wave theory, an individual closed system is described by a wavefunction ψ(q) and configuration q. The evolution of the wavefunction and configuration are respectively determined by the Schrödinger and guidance equations. The guidance equation states that the velocity field for the configuration is given by the quantum current divided by the density ∣ψ(q)∣2. We present the currents and associated guidance equations for any Hamiltonian given by a differential operator. These are derived directly from the Schrödinger equation, and also as Noether currents arising from a global phase symmetry associated with the wavefunction in configuration space.
Probability is an important question in the ontological interpretation of quantum mechanics. It has been discussed in some trajectory interpretations such as Bohmian mechanics and stochastic mechanics. New questions arise when the probability domain extends to the complex space, including the generation of complex trajectory, the definition of the complex probability, and the relation of the complex probability to the quantum probability. The complex treatment proposed in this article applies the optimal quantum guidance law to derive the stochastic differential equation governing a particle’s random motion in the complex plane. The probability distribution ρc(t,x,y) of the particle’s position over the complex plane z=x+iy is formed by an ensemble of the complex quantum random trajectories, which are solved from the complex stochastic differential equation. Meanwhile, the probability distribution ρc(t,x,y) is verified by the solution of the complex Fokker–Planck equation. It is shown that quantum probability |Ψ|2 and classical probability can be integrated under the framework of complex probability ρc(t,x,y), such that they can both be derived from ρc(t,x,y) by different statistical ways of collecting spatial points.
An interpretation of quantum mechanics is proposed which augments the stochastic pilot-wave model, introduced by Nelson [E. Nelson, Phys. Rev., 150, 1079 (1966)], with a dual guidance condition. Namely, in addition to the stochastic guidance condition which describes how the wave controls the particle (wave-to-particle guidance condition), we introduce another stochastic guidance condition describing how the particle guides the wave (particle-to-wave guidance condition). We therefore introduce an action-reaction principle in the pilot-wave formulation of quantum mechanics. The particle-to-wave guidance condition takes the form of spontaneous transitions of the wavefunction that are Poisson distributed in time. The wavefunction selected by the transition is influenced by its likelyhood with respect to the particle position, as well as certain conservation constraints. It is shown that the stochastic particle-to-wave guidance condition reproduces certain aspects of the spontaneous localization model of Ghirardi, Rimini and Weber [G. C. Ghirardi, A. Rimini and T. Weber, Phys. Rev. D, 34, 470 (1986)], and of Milburn's intrinsic decoherence model [G. J. Milburn, Phys. Rev. A, 44, 5401 (1991)], while avoiding some of the difficulties associated with them. The macroscopic limit of the dualist interpretation is analyzed for the case of a free body. Possible improvements of this interpretation are discussed.
Abstract We revisit the Bohr model of the atom through Brownian motion of the electron and the principles of stochastic optimal control. The electron is assumed to have a definite but random position, represented by a single real-valued stochastic process in physical space whose probability density obeys the Fokker–Planck (FP) equation. Because Brownian paths are not differentiable, the process carries two mean drifts, one for each direction of time. We treat the forward drift as the control field, while the backward drift is fixed by the density of the same process. The running cost of the control problem combines the two drifts into a time-symmetric kinetic term, and through the backward drift it inherits a dependence on the density, so the value becomes a functional on density space. Bellman’s dynamic-programming principle requires the control to minimize the expected action from every intermediate time and density onward. The drift therefore emerges as a feedback law on position and density, rather than from the global stationarity of a stochastic action. The resulting law-dependent HJB–FP system reduces to the Schrödinger equation once the standard single-valuedness condition is imposed on the reconstructed phase. For stationary hydrogen states the theory yields explicit drift fields in spherical coordinates and reproduces the standard radial and angular kinetic-energy averages of the quantum operator formalism. Direct trajectory-level simulations of the electron’s stochastic motion around the nucleus show the coordinate distributions converging to the Born marginals and the time-averaged energies reproducing the quantum expectation values. For the 2 p eigenstates with magnetic quantum number m = ± 1 , a phase-driven azimuthal drift makes the simulated trajectories circulate at the analytically predicted rate, and the angular momentum accumulated from the raw trajectory increments converges to exactly L z = m ℏ . The angular-momentum quantization postulated in the Bohr model thus reappears as a property of the simulated stochastic motion.
In the pilot-wave theory of quantum mechanics particles have definite positions and velocities and the system evolves deterministically. The velocity of a particle is determined by the wave function of the system (the guidance equation) and the wave function evolves according to Schrödinger’s equation. In this paper I first construct a Hamiltonian that gives Schrödinger’s equation and the guidance equation for the particle. I then find the Hamiltonian for a relativistic particle in Dirac’s theory and for a quantum scalar field.