Editing Double-slit experiment (section)

==Path-integral formulation==
[[Image:Wiener process 3d.png|thumb|200px|One of an infinite number of equally likely paths used in the Feynman path integral (see also: [[Wiener process]])]]

The double-slit experiment can illustrate the [[path integral formulation]] of quantum mechanics provided by Feynman.<ref>{{cite book|last=Zee |first=Anthony |title=Quantum Field Theory in a Nutshell |author-link=Anthony Zee |title-link=Quantum Field Theory in a Nutshell |publisher=Princeton University Press |edition=2nd |year=2010 |isbn=978-0-691-14034-6 |pages=2–16}}</ref> The path integral formulation replaces the classical notion of a single, unique trajectory for a system, with a sum over all possible trajectories.  The trajectories are added together by using [[functional integration]].

Each path is considered equally likely, and thus contributes the same amount.  However, the [[phase (waves)|phase]] of this contribution at any given point along the path is determined by the [[action (physics)|action]] along the path:
<math display="block">A_{\text{path}}(x,y,z,t) = e^{i S(x,y,z,t)}</math>

All these contributions are then added together, and the [[magnitude (mathematics)|magnitude]] of the final result is [[Square (algebra)|squared]], to get the probability distribution for the position of a particle:
<math display="block">p(x,y,z,t) \propto \left\vert \int_{\text{all paths}} e^{i S(x,y,z,t)} \right\vert ^2 </math>

As is always the case when calculating [[probability]], the results must then be [[Normalizing constant|normalized]] by imposing:
<math display="block">\iiint_{\text{all space}}p(x,y,z,t)\,dV = 1</math>

The probability distribution of the outcome is the normalized square of the norm of the [[superposition principle|superposition]], over all paths from the point of origin to the final point, of [[wave]]s [[wave propagation|propagating]] [[Proportionality (mathematics)|proportionally]] to the action along each path.  The differences in the cumulative action along the different paths (and thus the relative phases of the contributions) produces the [[Interference (wave propagation)|interference pattern]] observed by the double-slit experiment.  Feynman stressed that his formulation is merely a mathematical description, not an attempt to describe a real process that we can measure.