Editing Huygens–Fresnel principle (section)

==Generalized Huygens' principle==
Many books and references – e.g. (Greiner, 2002)<ref name="Greiner">{{cite book | author=Greiner W. | title= Quantum Electrodynamics | publisher=Springer, 2002}}</ref> and (Enders, 2009)<ref name="Enders">{{cite journal |last1=Enders |first1=Peter |title=Huygens' Principle as Universal Model of Propagation |journal=Latin-American Journal of Physics Education |date=2009 |volume=3 |issue=1 |pages=19–32 |url=https://lajpe.org/jan09/04_Peter_Enders.pdf }}</ref> - refer to the Generalized Huygens' Principle using the definition in ([[Feynman]], 1948).<ref name="Fey1">{{cite journal |last1=Feynman |first1=R. P. |title=Space-Time Approach to Non-Relativistic Quantum Mechanics |journal=Reviews of Modern Physics |date=1 April 1948 |volume=20 |issue=2 |pages=367–387 |doi=10.1103/RevModPhys.20.367 |bibcode=1948RvMP...20..367F |url=https://resolver.caltech.edu/CaltechAUTHORS:20140731-165931911 }}</ref>

Feynman defines the generalized principle in the following way:
{{bquote| "Actually Huygens’ principle is not correct in optics. It is replaced by Kirchoff’s [sic] modification which requires that both the amplitude and its derivative must be known on the adjacent surface. This is a consequence of the fact that the wave equation in optics is second order in the time. The wave equation of quantum mechanics is first order in the time; therefore, Huygens’ principle is correct for matter waves, action replacing time."}}

This clarifies the fact that in this context the generalized principle reflects the linearity of quantum mechanics and the fact that the quantum mechanics equations are first order in time. Finally only in this case the superposition principle fully apply, i.e. the wave function in a point P can be expanded as a superposition of waves on a border surface enclosing P. Wave functions can be interpreted in the usual quantum mechanical sense as probability densities where the formalism of [[Green's function (many-body theory)|Green's functions]] and [[propagator]]s apply. What is note-worthy is that this generalized principle is applicable for "matter waves" and not for light waves any more. The phase factor is now clarified as given by the [[Action (physics)|action]] and there is no more confusion why the phases of the wavelets are different from the one of the original wave and modified by the additional Fresnel parameters.

As per Greiner <ref name="Greiner"/> the generalized principle can be expressed for <math>t'>t </math> in the form:
:<math>\psi'(\mathbf{x}',t') = i \int d^3x \, G(\mathbf{x}',t';\mathbf{x},t)\psi(\mathbf{x},t)</math>
where ''G'' is the usual Green function that propagates in time the wave function <math>\psi</math>. This description resembles and generalize the initial Fresnel's formula of the classical model.