Editing Pilot wave theory (section)

===Mathematical formulation for a single particle===
The matter wave of de Broglie is described by the time-dependent Schrödinger equation:

:<math> i\, \hbar\, \frac{\,\partial \psi\,}{\partial t} = \left( - \frac{\hbar^2}{\,2m\,} \nabla^2 + V \right)\psi \quad</math>

The complex wave function can be represented as:

<math>\psi = \sqrt{\rho\,} \; \exp \left( \frac{i \, S}{\hbar} \right) ~</math>

By plugging this into the Schrödinger equation, one can derive two new equations for the real variables. The first is the [[Probability current#Continuity equation for quantum mechanics|continuity equation for the probability density]] <math>\,\rho\,:</math><ref name=Bohm1952a/>

:<math>\frac{\, \partial \rho \,}{\, \partial t \,} + \vec{\nabla} \cdot \left( \rho\, \vec{v} \right) = 0 ~ ,</math>

where the [[velocity field]] is determined by the “guidance equation”

:<math>\vec{v}\left(\,\vec{r},\,t\,\right) = \frac{1}{\,m\,} \, \vec{\nabla} S\left(\, \vec{r},\, t \,\right) ~ .</math>

According to pilot wave theory, the point particle and the matter wave are both real and distinct physical entities (unlike standard quantum mechanics, which postulates no physical particle or wave entities, only observed wave-particle duality). 
The pilot wave guides the motion of the point particles as described by the guidance equation.

Ordinary quantum mechanics and pilot wave theory are based on the same partial differential equation. The main difference is that in ordinary quantum mechanics, the Schrödinger equation is connected to reality by the Born postulate, which states that the probability density of the particle's position is given by <math>\; \rho = |\psi|^2 ~ .</math> Pilot wave theory considers the guidance equation to be the fundamental law, and sees the Born rule as a derived concept.

The second equation is a modified [[Hamilton–Jacobi equation]] for the action {{mvar|S}}:

:<math>- \frac{\partial S}{\partial t} = \frac{\;\left|\, \vec{\nabla} S \,\right|^2\,}{\,2m\,} + V + Q ~ ,</math>

where {{mvar|Q}} is the quantum potential defined by

:<math> Q = - \frac{\hbar^2}{\,2m\,} \frac{\nabla^2 \sqrt{\rho \,} }{\sqrt{ \rho \,} } ~.</math>

If we choose to neglect {{mvar|Q}}, our equation is reduced to the Hamilton–Jacobi equation of a classical point particle.{{efn|Strictly speaking, this is only a semiclassical limit;{{clarify|date=March 2012}} because the superposition principle still holds, one needs a “decoherence mechanism” to get rid of it. Interaction with the environment can provide this mechanism.}} So, the quantum potential is responsible for all the mysterious effects of quantum mechanics.

One can also combine the modified Hamilton–Jacobi equation with the guidance equation to derive a quasi-Newtonian equation of motion

:<math>m \, \frac{d}{dt} \, \vec{v} = - \vec{\nabla}( V + Q ) ~ ,</math>

where the hydrodynamic time derivative is defined as

:<math>\frac{d}{dt} = \frac{ \partial }{\, \partial t \,} + \vec{v} \cdot \vec{\nabla} ~ .</math>