Editing De Broglie–Bohm theory (section)

== Overview ==
De Broglie–Bohm theory is based on the following postulates:
* There is a configuration <math>q</math> of the universe, described by coordinates <math>q^k</math>, which is an element of the configuration space <math>Q</math>. The configuration space is different for different versions of pilot-wave theory. For example, this may be the space of positions <math>\mathbf{Q}_k</math> of <math>N</math> particles, or, in case of field theory, the space of field configurations <math>\phi(x)</math>. The configuration evolves (for spin=0) according to the guiding equation <math display="block">m_k \frac{d q^k}{dt}(t) = \hbar \nabla_k \operatorname{Im} \ln\psi(q,t) = \hbar \operatorname{Im}\left(\frac{\nabla_k \psi}{\psi}\right)(q, t) = \frac{m_k \mathbf{j}_k}{\psi^*\psi} = \operatorname{Re}\left(\frac{\mathbf{\hat P}_k\Psi}{\Psi}\right), </math> where <math>\mathbf{j}</math> is the [[probability current]] or probability flux, and <math>\mathbf{\hat P}</math> is the [[momentum operator]]. Here, <math>\psi(q,t)</math> is the standard complex-valued wavefunction from quantum theory, which evolves according to [[Schrödinger's equation]] <math display="block">i\hbar \frac{\partial}{\partial t}\psi(q,t) = - \sum_{i=1}^{N} \frac{\hbar^2}{2m_i} \nabla_i^2 \psi(q,t) + V(q)\psi(q,t).</math>This completes the specification of the theory for any quantum theory with Hamilton operator of type <math display="inline">H =\sum \frac{1}{2m_i}\hat{p}_i^2 + V(\hat{q})</math>.
* The configuration is distributed according to <math>|\psi(q,t)|^2</math> at some moment of time <math>t</math>, and this consequently holds for all times. Such a state is named quantum equilibrium. With quantum equilibrium, this theory agrees with the results of standard quantum mechanics.

Even though this latter relation is frequently presented as an axiom of the theory, Bohm presented it as derivable from statistical-mechanical arguments in the original papers of 1952. This argument was further supported by the work of Bohm in 1953 and was substantiated by Vigier and Bohm's paper of 1954, in which they introduced stochastic ''fluid fluctuations'' that drive a process of asymptotic relaxation from [[quantum non-equilibrium]] to quantum equilibrium (ρ → |ψ|<sup>2</sup>).<ref>Publications of D. Bohm in 1952 and 1953 and of J.-P. Vigier in 1954 as cited in {{Cite journal|author1=Antony Valentini|author2=Hans Westman|title=Dynamical origin of quantum probabilities |journal=Proc. R. Soc. A|year=2005|volume=461|number=2053|pages=253–272|doi=10.1098/rspa.2004.1394|bibcode=2005RSPSA.461..253V|arxiv = quant-ph/0403034 |citeseerx=10.1.1.252.849|s2cid=6589887}} [http://rspa.royalsocietypublishing.org/content/461/2053/253.full.pdf#page=3 p. 254].</ref>

=== Double-slit experiment ===
[[File:doppelspalt.svg|thumb|The Bohmian trajectories for an electron going through the two-slit experiment. A similar pattern was also extrapolated from [[weak measurement]]s of single photons.<ref>{{Cite journal |last1=Kocsis |first1=Sacha |last2=Braverman |first2=Boris |last3=Ravets |first3=Sylvain |last4=Stevens |first4=Martin J. |last5=Mirin |first5=Richard P. |last6=Shalm |first6=L. Krister |last7=Steinberg |first7=Aephraim M. |date=2011-06-03 |title=Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer |url=https://www.science.org/doi/10.1126/science.1202218 |journal=Science |language=en |volume=332 |issue=6034 |pages=1170–1173 |doi=10.1126/science.1202218 |pmid=21636767 |bibcode=2011Sci...332.1170K |s2cid=27351467 |issn=0036-8075}}</ref>]]

The [[double-slit experiment]] is an illustration of [[wave–particle duality]]. In it, a beam of particles (such as electrons) travels through a barrier that has two slits. If a detector screen is on the side beyond the barrier, the pattern of detected particles shows interference fringes characteristic of waves arriving at the screen from two sources (the two slits); however, the interference pattern is made up of individual dots corresponding to particles that had arrived on the screen. The system seems to exhibit the behaviour of both waves (interference patterns) and particles (dots on the screen).

If this experiment is modified so that one slit is closed, no interference pattern is observed. Thus, the state of both slits affects the final results. It can also be arranged to have a minimally invasive detector at one of the slits to detect which slit the particle went through. When that is done, the interference pattern disappears.<ref>{{Cite journal |last=Zeilinger |first=Anton |date=1999-03-01 |title=Experiment and the foundations of quantum physics |url=https://link.aps.org/doi/10.1103/RevModPhys.71.S288 |journal=Reviews of Modern Physics |language=en |volume=71 |issue=2 |pages=S288–S297 |doi=10.1103/RevModPhys.71.S288 |issn=0034-6861}}</ref>

In de Broglie–Bohm theory, the wavefunction is defined at both slits, but each particle has a well-defined trajectory that passes through exactly one of the slits. The final position of the particle on the detector screen and the slit through which the particle passes is determined by the initial position of the particle. Such initial position is not knowable or controllable by the experimenter, so there is an appearance of randomness in the pattern of detection. In Bohm's 1952 papers he used the wavefunction to construct a [[quantum potential]] that, when included in Newton's equations, gave the trajectories of the particles streaming through the two slits. In effect the wavefunction interferes with itself and guides the particles by the quantum potential in such a way that the particles avoid the regions in which the interference is destructive and are attracted to the regions in which the interference is constructive, resulting in the interference pattern on the detector screen.

To explain the behavior when the particle is detected to go through one slit, one needs to appreciate the role of the conditional wavefunction and how it results in the collapse of the wavefunction; this is explained below. The basic idea is that the environment registering the detection effectively separates the two wave packets in configuration space.