Editing Pilot wave theory (section)

===Derivation of the Schrödinger equation===
Pilot wave theory is based on [[Hamilton–Jacobi equation|Hamilton–Jacobi dynamics]],<ref>{{cite web |last=Towler |first=M. |date=10 February 2009 |title=De Broglie-Bohm pilot-wave theory and the foundations of quantum mechanics |url=http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html |publisher=University of Cambridge |access-date=2014-07-03 |archive-date=10 April 2016 |archive-url=https://web.archive.org/web/20160410173517/http://www.tcm.phy.cam.ac.uk/%7Emdt26/pilot_waves.html |url-status=dead }}</ref> rather than [[Lagrangian mechanics|Lagrangian]] or [[Hamiltonian dynamics]]. Using the Hamilton–Jacobi equation

:<math> H\left(\,\vec{x}\,, \;\vec{\nabla}_{\!x}\, S\,, \;t \,\right) + {\partial S \over \partial t}\left(\,\vec{x},\, t\,\right) = 0</math>

it is possible to derive the [[Schrödinger equation]]:

Consider a classical particle – the position of which is not known with certainty. We must deal with it statistically, so only the probability density <math>\rho (\vec{x},t)</math> is known. Probability must be conserved, i.e. <math>\int\rho\,\mathrm{d}^3\vec{x} = 1</math> for each <math>t</math>. Therefore, it must satisfy the continuity equation

:<math>\frac{\, \partial \rho \,}{ \partial t } = - \vec{\nabla} \cdot (\rho \,\vec{v} ) \qquad\qquad (1)</math>

where <math>\,\vec{v}(\vec{x},t)\,</math> is the velocity of the particle.

In the Hamilton–Jacobi formulation of [[classical mechanics]], velocity is given by <math>\; \vec{v}(\vec{x},t) = \frac{1}{\,m\,} \, \vec{\nabla}_{\!x} S(\vec{x},\,t) \;</math> where <math>\, S(\vec{x},t) \,</math> is a solution of the Hamilton-Jacobi equation

:<math>- \frac{\partial S}{\partial t} = \frac{\;\left|\,\nabla S\,\right|^2\,}{2m} + \tilde{V} \qquad\qquad (2)</math>

<math>\,(1)\,</math> and <math>\,(2)\,</math> can be combined into a single complex equation by introducing the complex function <math>\; \psi = \sqrt{\rho\,} \, e^\frac{\,i\,S\,}{\hbar} \;,</math> then the two equations are equivalent to

:<math>i\, \hbar\, \frac{\,\partial \psi\,}{\partial t} = \left( - \frac{\hbar^2}{2m} \nabla^2 +\tilde{V} - Q \right)\psi \quad</math>
with
:<math> \; Q = - \frac{\;\hbar^2\,}{\,2m\,} \frac{\nabla^2 \sqrt{\rho\,}}{\sqrt{\rho\,}}~.</math>

The time-dependent Schrödinger equation is obtained if we start with <math>\;\tilde{V} = V + Q \;,</math> the usual potential with an extra [[quantum potential]] <math>Q</math>. The quantum potential is the potential of the quantum force, which is proportional (in approximation) to the [[Curvature#Graph of a function|curvature]] of the amplitude of the wave function.

Note this potential is the same one that appears in the [[Madelung equations]], a classical analog of the Schrödinger equation.