BroglieBohm

The de Broglie–Bohm theory, also known as Bohmian mechanics or pilot-wave theory, is an interpretation of quantum mechanics that postulates that particles have definite positions at all times, guided by a real wave function.

In this framework, the wave function ψ evolves by the Schrödinger equation, and each particle i follows a deterministic trajectory given by dxi/dt = (ħ/m_i) Im(∇_i ψ/ψ) = (1/m_i) ∇_i S, where ψ = R e^{iS/ħ}. The wave function guides motion through a quantum potential Q = - (ħ^2/2m) ∇^2 R / R, producing interference and other quantum effects.

Ontology and predictions: The theory is a nonlocal hidden-variable theory. It reproduces standard quantum predictions for non-relativistic systems when the distribution of particle configurations matches the quantum equilibrium |ψ|^2. Measurement outcomes are outcomes of definite particle positions, without requiring wave function collapse.

History and status: Proposed by Louis de Broglie in 1925 and developed by David Bohm in 1952, it remains an actively studied interpretation. It is empirically equivalent to mainstream quantum mechanics for most experiments but differs in its explicit ontology and nonlocal dynamics, and faces challenges in relativistic and quantum-field theory contexts.

Theory in context: The de Broglie–Bohm approach has influenced discussions in the philosophy of quantum mechanics and is cited as an example of a deterministic hidden-variable interpretation that preserves realism, while confronting nonlocality and compatibility with relativity.