Editing De Broglie–Bohm theory (section)

== Derivations ==
De Broglie–Bohm theory has been derived many times and in many ways. Below are six derivations, all of which are very different and lead to different ways of understanding and extending this theory.
* [[Schrödinger equation#Derivation|Schrödinger's equation]] can be derived by using [[Photoelectric effect|Einstein's light quanta hypothesis]]: <math>E = \hbar \omega</math> and [[Matter wave|de Broglie's hypothesis]]: <math>\mathbf{p} = \hbar \mathbf{k}</math>.
:The guiding equation can be derived in a similar fashion. We assume a plane wave: <math>\psi(\mathbf{x}, t) = A e^{i(\mathbf{k}\cdot\mathbf{x} - \omega t)}</math>. Notice that <math>i\mathbf{k} = \nabla\psi/\psi</math>. Assuming that <math>\mathbf{p} = m \mathbf{v}</math> for the particle's actual velocity, we have that <math>\mathbf{v} = \frac{\hbar}{m} \operatorname{Im}\left(\frac{\nabla\psi}{\psi}\right)</math>. Thus, we have the guiding equation.
:Notice that this derivation does not use Schrödinger's equation.
* Preserving the density under the time evolution is another method of derivation. This is the method that Bell cites. It is this method that generalizes to many possible alternative theories. The starting point is the [[continuity equation]] <math>-\frac{\partial\rho}{\partial t} = \nabla \cdot (\rho v^\psi)</math> {{clarify|date=March 2018}} for the density <math>\rho = |\psi|^2</math>. This equation describes a probability flow along a current. We take the velocity field associated with this current as the velocity field whose integral curves yield the motion of the particle.
* A method applicable for particles without spin is to do a polar decomposition of the wavefunction and transform Schrödinger's equation into two coupled equations: the [[continuity equation]] from above and the [[Hamilton–Jacobi equation]]. This is the method used by Bohm in 1952. The decomposition and equations are as follows:

:Decomposition: <math>\psi(\mathbf{x}, t) = R(\mathbf{x}, t) e^{i S(\mathbf{x}, t) / \hbar}.</math> Note that <math>R^2(\mathbf{x}, t)</math> corresponds to the probability density <math>\rho(\mathbf{x}, t) = |\psi(\mathbf{x}, t)|^2</math>.

:Continuity equation: <math>-\frac{\partial\rho(\mathbf{x}, t)}{\partial t} = \nabla \cdot \left(\rho(\mathbf{x}, t) \frac{\nabla S(\mathbf{x}, t)}{m}\right)</math>.

:Hamilton–Jacobi equation: <math>\frac{\partial S(\mathbf{x}, t)}{\partial t} = -\left[\frac{1}{2m}(\nabla S(\mathbf{x}, t))^2 + V - \frac{\hbar ^2}{2m} \frac{\nabla^2R(\mathbf{x}, t)}{R(\mathbf{x}, t)}\right].</math>

:The Hamilton–Jacobi equation is the equation derived from a Newtonian system with potential <math>V - \frac{\hbar^2}{2m} \frac{\nabla^2 R}{R}</math> and velocity field <math>\frac{\nabla S}{m}.</math> The potential <math>V</math> is the classical potential that appears in Schrödinger's equation, and the other term involving <math>R</math> is the [[quantum potential]], terminology introduced by Bohm.

:This leads to viewing the quantum theory as particles moving under the classical force modified by a quantum force. However, unlike standard [[Newtonian mechanics]], the initial velocity field is already specified by <math>\frac{\nabla S}{m}</math>, which is a symptom of this being a first-order theory, not a second-order theory.
* A fourth derivation was given by Dürr et al.<ref name="dgz92" /> In their derivation, they derive the velocity field by demanding the appropriate transformation properties given by the various symmetries that Schrödinger's equation satisfies, once the wavefunction is suitably transformed. The guiding equation is what emerges from that analysis.
* A fifth derivation, given by Dürr et al.<ref name="dgtz04" /> is appropriate for generalization to quantum field theory and the Dirac equation. The idea is that a velocity field can also be understood as a first-order differential operator acting on functions. Thus, if we know how it acts on functions, we know what it is. Then given the Hamiltonian operator <math>H</math>, the equation to satisfy for all functions <math>f</math> (with associated multiplication operator <math>\hat{f}</math>) is <math>(v(f))(q) = \operatorname{Re}\frac{\left(\psi, \frac{i}{\hbar} [H, \hat f] \psi\right)}{(\psi, \psi)}(q)</math>, where <math>(v, w)</math> is the local Hermitian inner product on the value space of the wavefunction.

:This formulation allows for stochastic theories such as the creation and annihilation of particles.
* A further derivation has been given by Peter R. Holland, on which he bases his quantum-physics textbook ''The Quantum Theory of Motion''.<ref>Peter R. Holland: ''The quantum theory of motion'', Cambridge University Press, 1993 (re-printed 2000, transferred to digital printing 2004), {{ISBN|0-521-48543-6}}, p. 66 ff.</ref> It is based on three basic postulates and an additional fourth postulate that links the wavefunction to measurement probabilities:
*# A physical system consists in a spatiotemporally propagating wave and a point particle guided by it.
*# The wave is described mathematically by a solution <math>\psi</math> to Schrödinger's wave equation.
*# The particle motion is described by a solution to <math>\mathbf{\dot x}(t) = [\nabla S (\mathbf{x}(t), t))]/m</math> in dependence on initial condition <math>\mathbf{x}(t = 0)</math>, with <math>S</math> the phase of <math>\psi</math>.{{pb}}The fourth postulate is subsidiary yet consistent with the first three:
*# The probability <math>\rho(\mathbf{x}(t))</math> to find the particle in the differential volume <math>d^3 x</math> at time ''t'' equals <math>|\psi(\mathbf{x}(t))|^2</math>.