= Solution
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 $\Psi=R e^{iS/\hbar}$ using a dimensionful action $S$. Then
$$
\boxed{\dot{\mathbf Q}_i=\frac{\nabla_iS}{m_i}
=\frac\hbar{m_i}\operatorname{Im}\frac{\nabla_i\Psi}{\Psi}\bigg|_{\mathbf q=\mathbf Q}.}
$$
The 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,
$$
\partial_tS+\sum_i\frac{|\nabla_iS|^2}{2m_i}+V+Q=0,
\qquad Q=-\sum_i\frac{\hbar^2}{2m_i}\frac{\nabla_i^2R}{R}.
$$
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.<A Suggested Interpretation of the Quantum Theory in Terms of Hidden Variables I by David Bohm>[1952, first paper in the two-part interpretation]{ref}<A Suggested Interpretation of the Quantum Theory in Terms of Hidden Variables II by David Bohm>[1952, second paper on the measurement account]{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.
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