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David Bohm's 1952 formulation of de Broglie-Bohm theory supplements the wave function with definite particle positions guided by that wave function. For a spinless particle of mass $m$, writing $\psi =R{e}^{iS/\hslash }$ gives the guidance velocity $\nabla S/m$. The resulting trajectories and the Schrödinger equation together preserve the distribution $|\psi {|}^2$ when it holds initially, connecting the particle description with ordinary measurement statistics. His second paper examined the measurement process and its relation to the usual quantum mechanics predictions.^[ref][ref]

A physicist, Professor of theoretical physics at Birkbeck from 1961 to 1987.^[ref]

In de Broglie-Bohm theory, the system has both a wavefunction and an actual configuration of particle positions. The wavefunction obeys the Time-dependent Schrödinger equation; the configuration follows the guidance equation. For spinless particles without electromagnetic vector potentials, write $\Psi =R{e}^{iS/\hslash }$ using a dimensionful action $S$. ThenThe actual configuration selects a trajectory, while the guiding wave generally retains all branches. A measurement correlates system and apparatus configurations; the actual apparatus position lies in one recorded branch. Effective collapse follows when wave packets have separated sufficiently in configuration space, although the full wave continues its Schrödinger evolution.

Substituting the polar form into that equation yields the probability continuity equation and a modified Hamilton-Jacobi equation,The amplitude-dependent term is the quantum potential. Taking gradients gives a second-order force description, but initial velocities remain constrained by the guidance law. Nodes require care because the displayed quotient is singular there.

If the initial configuration distribution is $|\Psi {|}^2$, the same continuity equation preserves that distribution: quantum equilibrium equivariance. It then reproduces ordinary nonrelativistic quantum measurement statistics within the model's applicability. Equivariance preserves a distribution; it is not by itself a proof that every initial distribution equals $|\Psi {|}^2$.

For entangled particles the guiding wave lives on their joint configuration space, so one particle's velocity can depend on the distant actual configuration. This explicit nonlocality avoids the locality premise of a Bell inequality. In quantum equilibrium it does not permit controllable faster-than-light signalling. Spin extensions use a spinor wavefunction and its probability current; they need not assign a classical little spinning vector with simultaneous definite values for every spin component.

David Bohm developed this account in his two 1952 papers. The first sets out the dynamics and the second treats measurement and the relation to ordinary predictions.^[ref][ref] The theory supplies additional dynamical variables and a particular nonlocal account of outcomes, rather than deriving a unique interpretation from the standard wavefunction alone.

In Bohmian mechanics, the corpuscles supply an actual configuration of matter, not a further branch of the wave function. Their positions and the guiding wave together specify the physical state. In a common nonrelativistic spinless formulation, the wave obeys the Schrödinger equation while the actual configuration $Q=(Q_1,\dots ,Q_N)$ obeyswhere the expression is applied away from nodes of $\Psi$. These are coupled in the sense that the wave guides the configuration; in the standard theory the wave's Schrödinger evolution does not receive a reciprocal source term from the actual particle positions. This is not a model of classical little balls carrying every ordinary observable as a pre-existing intrinsic attribute.

The point of adding a configuration is especially clear in a measurement. A wave may evolve into several macroscopically separated pointer packets. The apparatus's actual particle configuration lies in one packet, so there is a definite pointer record even though the full wave has not undergone fundamental collapse. The corpuscles thereby connect the mathematical description with the location of actual apparatus, records and observers. David Bohm's original papers provide a theory of individual processes and analyse measurement in this spirit.^[ref][ref]

For a subsystem, inserting the actual environment configuration into the universal wave gives a conditional wave function, schematically $\psi (x,t)\propto \Psi (x,Y(t),t)$. If the branches are suitably disjoint in the environment's configuration coordinates and remain dynamically isolated, the occupied branch supplies an effective collapse in Bohmian mechanics. The unoccupied branches are not simply erased by definition: before permanent separation, wave packets can recombine and affect interference. The conditional and effective wave-function constructions make this distinction precise.^[ref]

The Born rule also needs an account. The guidance current is $j_k=(\hslash /m_k)\mathrm{Im}({\Psi }^{\ast }{\nabla }_k\Psi )$, and both $|\Psi {|}^2$ and an ensemble of guided configurations satisfy the associated continuity equation. Hence an initial quantum equilibrium distribution $\rho =|\Psi {|}^2$ is preserved. Equivariance supports the ordinary statistical predictions, but the guidance law alone does not prove that an arbitrarily prepared ensemble has that distribution.

The corpuscles are therefore neither detectors that a conscious observer adds to the theory nor dispensable labels attached after the statistics have been calculated. They constitute a primitive ontology that provides definite configurations, trajectories and records. Whether this is preferable to a wave-only ontology is a further explanatory judgement; its empirical equivalence in quantum equilibrium does not make the two accounts identical in their claims about what exists.

Bohmian mechanics retains definite particle positions and deterministic trajectories, which makes it resemble classical mechanics at one level. But the guiding dynamics depends on a wavefunction obeying the Schrödinger equation, with velocity schematically ${\dot{Q}}_i=(\hslash /m_i)\mathrm{Im}({\nabla }_i\psi /\psi )(Q)$. For many particles this wavefunction is defined on configuration space and can be entangled. The velocity of one particle can consequently depend on the configuration of distant particles. Classical-looking trajectories therefore do not entail classical locality or separability.

Writing $\psi =R{e}^{iS/\hslash }$ also yields a Hamilton-Jacobi-like equation with quantum potential $Q=-{\sum }_i{\hslash }^2{\nabla }_i^{2}R/(2m_{i}R)$. The quantum potential is not generally a sum of ordinary pair potentials in three-space; its configuration dependence carries the nonclassical correlations. A stationary real bound-state wavefunction can give a stationary particle configuration, unlike the orbit one might infer from a classical Coulomb picture. Measurement outcomes are contextual: the actual apparatus interaction helps determine which position record is produced.

With the quantum-equilibrium distribution $\rho =|\psi {|}^2$, the guidance and wave equations preserve that distribution and reproduce the usual quantum statistics. Determinism alone does not establish that distribution; equilibrium or typicality is an additional explanatory component. The theory also leaves philosophical choices about whether the wavefunction is a physical field, a law-like object or part of a broader ontology.

The significant departure is therefore not simply replacing chance by hidden classical motion. It is the entangled guiding structure, nonlocal dependence and contextual relation between measurements and properties. The retained particles make the account continuous with some classical intuitions, while its actual law and ontology depart substantially from classical many-body mechanics.^[ref][ref]

For a measurement specifically, the apparatus configuration occupies one region associated with a record. The guidance law evolves that actual configuration into one branch of the entangled system-apparatus wavefunction; disjoint unoccupied branches do not contain the actual pointer. In quantum equilibrium the distribution of those occupied outcome regions is the Born distribution. Thus the account supplies a definite record without requiring stochastic wavefunction collapse, while paying the costs of added configuration ontology, nonlocal guidance and a justified equilibrium measure.

Bohmian mechanics retains definite particle positions and deterministic trajectories, which makes it resemble classical mechanics at one level. But the guiding dynamics depends on a wavefunction obeying the Schrödinger equation, with velocity schematically ${\dot{Q}}_i=(\hslash /m_i)\mathrm{Im}({\nabla }_i\psi /\psi )(Q)$. For many particles this wavefunction is defined on configuration space and can be entangled. The velocity of one particle can consequently depend on the configuration of distant particles. Classical-looking trajectories therefore do not entail classical locality or separability.

Writing $\psi =R{e}^{iS/\hslash }$ also yields a Hamilton-Jacobi-like equation with quantum potential $Q=-{\sum }_i{\hslash }^2{\nabla }_i^{2}R/(2m_{i}R)$. The quantum potential is not generally a sum of ordinary pair potentials in three-space; its configuration dependence carries the nonclassical correlations. A stationary real bound-state wavefunction can give a stationary particle configuration, unlike the orbit one might infer from a classical Coulomb picture. Measurement outcomes are contextual: the actual apparatus interaction helps determine which position record is produced.

With the quantum-equilibrium distribution $\rho =|\psi {|}^2$, the guidance and wave equations preserve that distribution and reproduce the usual quantum statistics. Determinism alone does not establish that distribution; equilibrium or typicality is an additional explanatory component. The theory also leaves philosophical choices about whether the wavefunction is a physical field, a law-like object or part of a broader ontology.

The significant departure is therefore not simply replacing chance by hidden classical motion. It is the entangled guiding structure, nonlocal dependence and contextual relation between measurements and properties. The retained particles make the account continuous with some classical intuitions, while its actual law and ontology depart substantially from classical many-body mechanics.^[ref][ref]