• Bohmian Quantum Mechanics

Bohmian mechanics, which is also called the de Broglie-Bohm theory, the pilot-wave model, and the causal interpretation of quantum mechanics, is a version of quantum theory discovered by Louis de Broglie in 1927 and rediscovered by David Bohm in 1952. In Bohmian mechanics a system of particles is described in part by its wave function, evolving according to Schrödinger's equation. However, the wave function provides only a partial description of the system. This description is completed by the specification of the actual positions of the particles. The latter evolves according to the “guiding equation,” which expresses the velocities of the particles in terms of the wave function. Thus, in Bohmian mechanics the configuration of a system of particles evolves via a deterministic motion choreographed by the wave function. In particular, when a particle is sent into a two-slit apparatus, the slit through which it passes and its location upon arrival on the photographic plate are completely determined by its initial position and wave function.

Bohmian mechanics can be presented as a first-order or a second-order theory. In the first-order presentation the velocity, the rate of change of position, is fundamental. It is this quantity, given by the guiding equation, that the theory specifies directly and simply. The second-order (Newtonian) concepts of acceleration and force, work ,and energy do not play any fundamental role. Bohm, however, did not regard his theory in this way. Rather, he regarded it, fundamentally, as a second-order theory, describing particles moving under the influence of forces, among which, however, is a force stemming from a “quantum potential.”

In his 1952 hidden-variables paper, Bohm arrived at his theory by writing the wave function in the polar form ψ = Rexp(iS/ℏ), where S and R are real, with R nonnegative, and rewriting Schrödinger's equation in terms of these new variables to obtain a pair of coupled evolution equations: the continuity equation for R² and a modified Hamilton-Jacobi equation for S. This differs from the usual classical Hamilton-Jacobi equation only by the appearance of an extra term, quantum potential, alongside the classical potential energy term. Bohm then used the modified Hamilton-Jacobi equation to define particle trajectories just as one does for the classical Hamilton-Jacobi equation.