Editing Photon (section)

== Wave–particle duality and uncertainty principles ==

Photons obey the laws of quantum mechanics, and so their behavior has both wave-like and particle-like aspects. When a photon is detected by a measuring instrument, it is registered as a single, particulate unit. However, the ''probability'' of detecting a photon is calculated by equations that describe waves. This combination of aspects is known as [[wave–particle duality]]. For example, the [[probability distribution]] for the location at which a photon might be detected displays clearly wave-like phenomena such as [[diffraction]] and [[Interference (wave propagation)|interference]]. A single photon passing through a [[double-slit experiment|double slit]] has its energy received at a point on the screen with a probability distribution given by its interference pattern determined by [[Maxwell's equations|Maxwell's wave equations]].<ref name="Taylor1909">
{{cite journal
 | first1=G. I. |last1=Taylor | authorlink1=Geoffrey Ingram Taylor
 | title=Interference Fringes with Feeble Light
 | journal = [[Mathematical Proceedings of the Cambridge Philosophical Society]]
 | volume=15
 | page=114
 | date=1909
 | url=https://archive.org/details/proceedingsofcam15190810camb/page/114/mode/2up | access-date=7 December 2024
}}</ref> However, experiments confirm that the photon is ''not'' a short pulse of electromagnetic radiation; a photon's Maxwell waves will diffract, but photon energy does not spread out as it propagates, nor does this energy divide when it encounters a [[beam splitter]].<ref name="Saleh">{{cite book |last1=Saleh |first1=B. E. A. |title=Fundamentals of Photonics |last2=Teich |first2=M. C. |publisher=Wiley |year=2007 |isbn=978-0-471-35832-9 |language=en |name-list-style=amp}}</ref> Rather, the received photon acts like a [[point-like particle]] since it is absorbed or emitted ''as a whole'' by arbitrarily small systems, including systems much smaller than its wavelength, such as an atomic nucleus (≈10<sup>−15</sup> m across) or even the point-like [[electron]].

While many introductory texts treat photons using the mathematical techniques of non-relativistic quantum mechanics, this is in some ways an awkward oversimplification, as photons are by nature intrinsically relativistic. Because photons have zero [[rest mass]], no [[wave function]] defined for a photon can have all the properties familiar from wave functions in non-relativistic quantum mechanics.{{efn|The issue was first formulated by Theodore Duddell Newton and [[Eugene Wigner]].<ref>{{cite journal|last1=Newton|first1=T.D.|last2=Wigner|first2=E.P.|author-link2=Eugene Wigner|year=1949|title=Localized states for elementary particles|journal=[[Reviews of Modern Physics]]|volume=21|pages=400–406|doi=10.1103/RevModPhys.21.400|bibcode=1949RvMP...21..400N|issue=3|url=https://cds.cern.ch/record/1062641/files/RevModPhys.21.400.pdf|doi-access=free|access-date=2023-06-21|archive-date=2023-05-16|archive-url=https://web.archive.org/web/20230516123629/http://cds.cern.ch/record/1062641/files/RevModPhys.21.400.pdf|url-status=live}}</ref><ref>{{cite journal|last=Bialynicki-Birula|first=I.|year=1994|title=On the wave function of the photon|journal=[[Acta Physica Polonica A]]|volume=86|issue=1–2|pages=97–116|doi=10.12693/APhysPolA.86.97|bibcode=1994AcPPA..86...97B|doi-access=free}}</ref><ref>{{cite journal|last=Sipe|first=J.E.|year=1995|title=Photon wave functions|journal=Physical Review A|volume=52|pages=1875–1883|doi=10.1103/PhysRevA.52.1875|pmid=9912446|bibcode=1995PhRvA..52.1875S|issue=3}}</ref> The challenges arise from the fundamental nature of the [[Lorentz group]], which describes the symmetries of [[spacetime]] in special relativity. Unlike the generators of [[Galilean transformation]]s, the generators of [[Lorentz boost]]s do not commute, and so simultaneously assigning low uncertainties to all coordinates of a relativistic particle's position becomes problematic.<ref>{{cite book|last=Bialynicki-Birula|first=I.|year=1996|title=V Photon Wave Function|journal=Progess in Optics|volume=36|pages=245–294|doi=10.1016/S0079-6638(08)70316-0|series=[[Progress in Optics]]|isbn=978-0-444-82530-8|bibcode=1996PrOpt..36..245B|s2cid=17695022 }}</ref>}} In order to avoid these difficulties, physicists employ the second-quantized theory of photons described below, [[quantum electrodynamics]], in which photons are quantized excitations of electromagnetic modes.<ref name="scully1997">{{cite book |last1=Scully |first1=M. O. |url=https://books.google.com/books?id=20ISsQCKKmQC |title=Quantum Optics |last2=Zubairy |first2=M. S. |publisher=Cambridge University Press |year=1997 |isbn=978-0-521-43595-6 |location=Cambridge, England |language=en-uk |access-date=2016-10-06 |archive-date=2024-05-13 |archive-url=https://web.archive.org/web/20240513012544/https://books.google.com/books?id=20ISsQCKKmQC |url-status=live }}</ref>

Another difficulty is finding the proper analogue for the [[uncertainty principle]], an idea frequently attributed to Heisenberg, who introduced the concept in analyzing a [[thought experiment]] involving [[Heisenberg's microscope|an electron and a high-energy photon]]. However, Heisenberg did not give precise mathematical definitions of what the "uncertainty" in these measurements meant. The precise mathematical statement of the position–momentum uncertainty principle is due to [[Earle Hesse Kennard|Kennard]], [[Wolfgang Pauli|Pauli]], and [[Hermann Weyl|Weyl]].<ref>{{Cite journal|last1=Busch|first1=Paul|author-link1=Paul Busch (physicist) |last2=Lahti|first2=Pekka|last3=Werner|first3=Reinhard F.|date=2013-10-17|title=Proof of Heisenberg's Error-Disturbance Relation|journal=Physical Review Letters|language=en|volume=111|issue=16|pages=160405|doi=10.1103/PhysRevLett.111.160405|pmid=24182239|arxiv=1306.1565|bibcode=2013PhRvL.111p0405B|s2cid=24507489|issn=0031-9007|url=https://www.repo.uni-hannover.de/bitstream/123456789/8834/1/Proof%20of%20Heisenberg%e2%80%99s%20Error-Disturbance%20Relation.pdf}}</ref><ref>{{Cite journal|last=Appleby|first=David Marcus|date=2016-05-06|title=Quantum Errors and Disturbances: Response to Busch, Lahti and Werner|journal=Entropy|language=en|volume=18|issue=5|pages=174|doi=10.3390/e18050174|arxiv=1602.09002|bibcode=2016Entrp..18..174A|doi-access=free}}</ref> The uncertainty principle applies to situations where an experimenter has a choice of measuring either one of two "canonically conjugate" quantities, like the position and the momentum of a particle. According to the uncertainty principle, no matter how the particle is prepared, it is not possible to make a precise prediction for both of the two alternative measurements: if the outcome of the position measurement is made more certain, the outcome of the momentum measurement becomes less so, and vice versa.<ref name="L&L">{{cite book |last1=Landau |first1=Lev D. |url=https://archive.org/details/QuantumMechanics_104 |title=Quantum Mechanics: Non-Relativistic Theory |last2=Lifschitz |first2=Evgeny M. |publisher=[[Pergamon Press]] |year=1977 |isbn=978-0-08-020940-1 |edition=3rd |volume=3 |language=en |oclc=2284121 |author-link1=Lev Landau |author-link2=Evgeny Lifshitz}}</ref> A [[coherent state]] minimizes the overall uncertainty as far as quantum mechanics allows.<ref name="scully1997"/> [[Quantum optics]] makes use of coherent states for modes of the electromagnetic field. There is a tradeoff, reminiscent of the position–momentum uncertainty relation, between measurements of an electromagnetic wave's amplitude and its phase.<ref name="scully1997"/> This is sometimes informally expressed in terms of the uncertainty in the number of photons present in the electromagnetic wave, <math>\Delta N</math>, and the uncertainty in the phase of the wave, <math>\Delta \phi</math>. However, this cannot be an uncertainty relation of the Kennard–Pauli–Weyl type, since unlike position and momentum, the phase <math>\phi</math> cannot be represented by a [[Hermitian operator]].<ref>{{Cite journal |last1=Busch |first1=P. |last2=Grabowski |first2=M. |last3=Lahti |first3=P. J. |date=January 1995 |title=Who Is Afraid of POV Measures? Unified Approach to Quantum Phase Observables |journal=[[Annals of Physics]] |language=en |volume=237 |issue=1 |pages=1–11 |bibcode=1995AnPhy.237....1B |doi=10.1006/aphy.1995.1001}}</ref>