Editing Uncertainty principle (section)

== History ==
{{See also|History of quantum mechanics}}
In 1925 Heisenberg published the [[Umdeutung paper|''Umdeutung'' (reinterpretation) paper]] where he showed that central aspect of quantum theory was the non-[[commutativity]]: the theory implied that the relative order of position and momentum measurement was significant. Working with [[Max Born]] and [[Pascual Jordan]], he continued to develop [[matrix mechanics]], that would become the first modern quantum mechanics formulation.<ref>{{Cite book |last=Whittaker |first=Edmund T. |title=A history of the theories of aether & electricity|volume= II: The modern theories, 1900–1926 |date=1989 |publisher=Dover Publ |isbn=978-0-486-26126-3 |edition=Repr |location=New York|page=267}}</ref> 
[[File:Heisenbergbohr.jpg|thumb|Werner Heisenberg and Niels Bohr]]
In March 1926, working in Bohr's institute, Heisenberg realized that the non-[[commutativity]] implies the uncertainty principle. Writing to [[Wolfgang Pauli]] in February 1927, he worked out the basic concepts.<ref>{{Cite web |title=This Month in Physics History |url=http://www.aps.org/publications/apsnews/200802/physicshistory.cfm |access-date=2023-11-04 |website=www.aps.org |language=en |archive-date=2011-01-30 |archive-url=https://web.archive.org/web/20110130195156/http://aps.org/publications/apsnews/200802/physicshistory.cfm |url-status=live }}</ref>

In his celebrated 1927 paper "{{lang|de|Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik}}" ("On the Perceptual Content of Quantum Theoretical Kinematics and Mechanics"), Heisenberg established this expression as the minimum amount of unavoidable momentum disturbance caused by any position measurement,<ref name=":0" /> but he did not give a precise definition for the uncertainties Δx and Δ''p''. Instead, he gave some plausible estimates in each case separately. His paper gave an analysis in terms of a microscope that Bohr showed was incorrect; Heisenberg included an addendum to the publication.

In his 1930 Chicago lecture<ref name="Heisenberg_1930">{{Citation |first=W. |last=Heisenberg |year=1930 |title=Physikalische Prinzipien der Quantentheorie |language=de|location=Leipzig |publisher=Hirzel }} English translation ''The Physical Principles of Quantum Theory''. Chicago: University of Chicago Press, 1930.</ref> he refined his principle:
{{NumBlk|:|<math>\Delta x \, \Delta p\gtrsim h</math>|{{EquationRef|A1}}}}

Later work broadened the concept. Any two variables that do not commute cannot be measured simultaneously—the more precisely one is known, the less precisely the other can be known. Heisenberg wrote:<blockquote>It can be expressed in its simplest form as follows: One can never know with perfect accuracy both of those two important factors which determine the movement of one of the smallest particles—its position and its velocity. It is impossible to determine accurately ''both'' the position and the direction and speed of a particle ''at the same instant''.<ref>Heisenberg, W., ''Die Physik der Atomkerne'', Taylor & Francis, 1952, p. 30.</ref></blockquote>

[[Earle Hesse Kennard|Kennard]]<ref name="Kennard" /><ref name=Sen2014 />{{rp|204}} in 1927 first proved the modern inequality:
{{NumBlk|:|<math>\sigma_x\sigma_p\ge\frac{\hbar}{2}</math>|{{EquationRef|A2}}}}
where {{math|1=''ħ'' = {{sfrac|''h''|2''π''}}}}, and {{math|''σ<sub>x</sub>''}}, {{math|''σ<sub>p</sub>''}} are the standard deviations of position and momentum. (Heisenberg only proved relation ({{EquationNote|A2}}) for the special case of Gaussian states.<ref name="Heisenberg_1930"/>) In 1929 Robertson generalized the inequality to all observables and in 1930 Schrödinger extended the form to allow non-zero covariance of the operators; this result is referred to as Robertson-Schrödinger inequality.<ref name=Sen2014 />{{rp|204}}

=== Terminology and translation ===
Throughout the main body of his original 1927 paper, written in German, Heisenberg used the word "Ungenauigkeit",<ref name=":0" />
to describe the basic theoretical principle. Only in the endnote did he switch to the word "Unsicherheit". Later on, he always used "Unbestimmtheit". When the English-language version of Heisenberg's textbook, ''The Physical Principles of the Quantum Theory'', was published in 1930, however, only the English word "uncertainty" was used, and it became the term in the English language.<ref>{{Citation |first1=David |last1=Cassidy |year=2009 |title=Beyond Uncertainty: Heisenberg, Quantum Physics, and the Bomb |location= New York |publisher=Bellevue Literary Press |page=185 |bibcode=2010PhT....63a..49C |bibcode-access=free |last2=Saperstein |first2=Alvin M. |volume=63 |issue=1 |journal=Physics Today |doi=10.1063/1.3293416 |doi-access=free }}</ref>

=== Heisenberg's microscope ===
[[File:Heisenberg gamma ray microscope.svg|thumb|200px|right|Heisenberg's gamma-ray microscope for locating an electron (shown in blue). The incoming gamma ray (shown in green) is scattered by the electron up into the microscope's aperture angle ''θ''. The scattered gamma-ray is shown in red. Classical [[optics]] shows that the electron position can be resolved only up to an uncertainty Δ''x'' that depends on ''θ'' and the wavelength ''λ'' of the incoming light.]]
{{Main article|Heisenberg's microscope}}

The principle is quite counter-intuitive, so the early students of quantum theory had to be reassured that naive measurements to violate it were bound always to be unworkable. One way in which Heisenberg originally illustrated the intrinsic impossibility of violating the uncertainty principle is by using the [[observer effect (physics)|observer effect]] of an imaginary microscope as a measuring device.<ref name="Heisenberg_1930"/>

He imagines an experimenter trying to measure the position and momentum of an [[electron]] by shooting a [[photon]] at it.<ref name=GreensteinZajonc2006>{{cite book|first1=George |last1=Greenstein|first2=Arthur |last2=Zajonc|authorlink2=Arthur Zajonc|title=The Quantum Challenge: Modern Research on the Foundations of Quantum Mechanics|year=2006|publisher=Jones & Bartlett Learning|isbn=978-0-7637-2470-2}}</ref>{{rp|49–50}}
* Problem 1 – If the photon has a short [[wavelength]], and therefore, a large momentum, the position can be measured accurately. But the photon scatters in a random direction, transferring a large and uncertain amount of momentum to the electron. If the photon has a long [[wavelength]] and low momentum, the collision does not disturb the electron's momentum very much, but the scattering will reveal its position only vaguely.
* Problem 2 – If a large [[aperture]] is used for the microscope, the electron's location can be well resolved (see [[Angular resolution#The_Rayleigh_criterion|Rayleigh criterion]]); but by the principle of [[conservation of momentum]], the transverse momentum of the incoming photon affects the electron's beamline momentum and hence, the new momentum of the electron resolves poorly. If a small aperture is used, the accuracy of both resolutions is the other way around.

The combination of these trade-offs implies that no matter what photon wavelength and aperture size are used, the product of the uncertainty in measured position and measured momentum is greater than or equal to a lower limit, which is (up to a small numerical factor) equal to the [[Planck constant]].<ref>{{Citation |last1=Tipler |first1=Paul A. |first2=Ralph A. |last2=Llewellyn |title=Modern Physics |volume=3 |publisher=W.H. Freeman & Co. |year=1999 |isbn=978-1572591646|lccn= 98046099
|url-access=|url=https://archive.org/details/modernphysics0003tipl |page=3 }}</ref> Heisenberg did not care to formulate the uncertainty principle as an exact limit, and preferred to use it instead, as a heuristic quantitative statement, correct up to small numerical factors, which makes the radically new noncommutativity of quantum mechanics inevitable.

===Intrinsic quantum uncertainty===

Historically, the uncertainty principle has been confused<ref>{{Citation|last=Furuta|first=Aya|title=One Thing Is Certain: Heisenberg's Uncertainty Principle Is Not Dead|journal=Scientific American|year=2012|url=https://www.scientificamerican.com/article/heisenbergs-uncertainty-principle-is-not-dead/|access-date=2018-10-20|archive-date=2022-04-01|archive-url=https://web.archive.org/web/20220401183444/https://www.scientificamerican.com/article/heisenbergs-uncertainty-principle-is-not-dead/|url-status=live}}</ref><ref name="Ozawa2003">{{Citation | last=Ozawa | first=Masanao | title=Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement | journal=Physical Review A | volume=67 | year=2003 | doi=10.1103/PhysRevA.67.042105|arxiv = quant-ph/0207121 |bibcode = 2003PhRvA..67d2105O | issue=4 | pages=42105 | s2cid=42012188}}</ref> with a related effect in [[physics]], called the [[observer effect (physics)|observer effect]], which notes that measurements of certain systems cannot be made without affecting the system,<ref>{{Citation |last=Wheeler |first=John Archibald |title=The 'Past' and the 'Delayed-Choice' Double-Slit Experiment |date=1978-01-01 |url=https://www.sciencedirect.com/science/article/pii/B9780124732506500066 |work=Mathematical Foundations of Quantum Theory |pages=9–48 |editor-last=Marlow |editor-first=A. R. |access-date=2023-07-19 |publisher=Academic Press |language=en |doi=10.1016/b978-0-12-473250-6.50006-6 |isbn=978-0-12-473250-6 |archive-date=2022-12-10 |archive-url=https://web.archive.org/web/20221210014455/https://www.sciencedirect.com/science/article/pii/B9780124732506500066 |url-status=live }}</ref><ref>{{Citation |last=Wheeler |first=John Archibald |title=Include the Observer in the Wave Function? |date=1977 |url=https://doi.org/10.1007/978-94-010-1196-9_1 |work=Quantum Mechanics, A Half Century Later: Papers of a Colloquium on Fifty Years of Quantum Mechanics, Held at the University Louis Pasteur, Strasbourg, May 2–4, 1974 |pages=1–18 |editor-last=Lopes |editor-first=José Leite |access-date=2023-07-19 |series=Episteme |place=Dordrecht |publisher=Springer Netherlands |language=en |doi=10.1007/978-94-010-1196-9_1 |isbn=978-94-010-1196-9 |editor2-last=Paty |editor2-first=Michel |archive-date=2024-02-23 |archive-url=https://web.archive.org/web/20240223170245/https://link.springer.com/chapter/10.1007/978-94-010-1196-9_1 |url-status=live }}</ref> that is, without changing something in a system. Heisenberg used such an observer effect at the quantum level (see below) as a physical "explanation" of quantum uncertainty.<ref>Werner Heisenberg, ''The Physical Principles of the Quantum Theory'', p. 20</ref> It has since become clearer, however, that the uncertainty principle is inherent in the properties of all [[wave|wave-like systems]],<ref name="Rozema">{{Cite journal | last1 = Rozema | first1 = L. A. | last2 = Darabi | first2 = A. | last3 = Mahler | first3 = D. H. | last4 = Hayat | first4 = A. | last5 = Soudagar | first5 = Y. | last6 = Steinberg | first6 = A. M. | doi = 10.1103/PhysRevLett.109.100404 |arxiv = 1208.0034v2| title = Violation of Heisenberg's Measurement–Disturbance Relationship by Weak Measurements | journal = Physical Review Letters | volume = 109 | issue = 10 | year = 2012 | pmid = 23005268|bibcode = 2012PhRvL.109j0404R | page=100404| s2cid = 37576344 }}</ref> and that it arises in quantum mechanics simply due to the [[matter wave]] nature of all quantum objects.<ref>{{Cite journal |last=De Broglie |first=Louis |date=October 1923 |title=Waves and Quanta |journal=Nature |language=en |volume=112 |issue=2815 |pages=540 |doi=10.1038/112540a0 |bibcode=1923Natur.112..540D |s2cid=186242764 |issn=1476-4687|doi-access=free }}</ref> Thus, the uncertainty principle actually states a fundamental property of quantum systems and is not a statement about the observational success of current technology.<ref name=nptel>{{YouTube|TcmGYe39XG0|Indian Institute of Technology Madras, Professor V. Balakrishnan, Lecture 1 – Introduction to Quantum Physics; Heisenberg's uncertainty principle, National Programme of Technology Enhanced Learning}}</ref>