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In physics, complementarity is a conceptual aspect of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
that Niels Bohr regarded as an essential feature of the theory. The complementarity principle holds that objects have certain pairs of complementary properties which cannot all be observed or measured simultaneously. An example of such a pair is position and momentum. Bohr considered one of the foundational truths of quantum mechanics to be the fact that setting up an experiment to measure one quantity of a pair, for instance the position of an
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have n ...
, excludes the possibility of measuring the other, yet understanding both experiments is necessary to characterize the object under study. In Bohr's view, the behavior of atomic and subatomic objects cannot be separated from the measuring instruments that create the context in which the measured objects behave. Consequently, there is no "single picture" that unifies the results obtained in these different experimental contexts, and only the "totality of the phenomena" together can provide a completely informative description.


History

Niels Bohr apparently conceived of the principle of complementarity during a skiing vacation in Norway in February and March 1927, during which he received a letter from
Werner Heisenberg Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a Über quantentheoretische Umdeutung kinematis ...
regarding an as-yet-unpublished result, a thought experiment about a microscope using gamma rays. This thought experiment implied a tradeoff between uncertainties that would later be formalized as the
uncertainty principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
. To Bohr, Heisenberg's paper did not make clear the distinction between a position measurement merely disturbing the momentum value that a particle carried and the more radical idea that momentum was meaningless or undefinable in a context where position was measured instead. Upon returning from his vacation, by which time Heisenberg had already submitted his paper for publication, Bohr convinced Heisenberg that the uncertainty tradeoff was a manifestation of the deeper concept of complementarity. Heisenberg duly appended a note to this effect to his paper, before its publication, stating:
Bohr has brought to my attention hatthe uncertainty in our observation does not arise exclusively from the occurrence of discontinuities, but is tied directly to the demand that we ascribe equal validity to the quite different experiments which show up in the articulatetheory on one hand, and in the wave theory on the other hand.
Bohr publicly introduced the principle of complementarity in a lecture he delivered on 16 September 1927 at the International Physics Congress held in
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, attended by most of the leading physicists of the era, with the notable exceptions of
Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born Theoretical physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for d ...
, Schrödinger, and Dirac. However, these three were in attendance one month later when Bohr again presented the principle at the Fifth Solvay Congress in
Brussels, Belgium Brussels (french: Bruxelles or ; nl, Brussel ), officially the Brussels-Capital Region (All text and all but one graphic show the English name as Brussels-Capital Region.) (french: link=no, Région de Bruxelles-Capitale; nl, link=no, Bruss ...
. The lecture was published in the proceedings of both of these conferences, and was republished the following year in ''Naturwissenschaften'' (in German) and in ''Nature'' (in English). Available in the collection of Bohr's early writings, ''Atomic Theory and the Description of Nature'' (1934). In his original lecture on the topic, Bohr pointed out that just as the finitude of the speed of light implies the impossibility of a sharp separation between space and time (relativity), the finitude of the quantum of action implies the impossibility of a sharp separation between the behavior of a system and its interaction with the measuring instruments and leads to the well-known difficulties with the concept of 'state' in quantum theory; the notion of complementarity is intended to capture this new situation in epistemology created by quantum theory. Physicists F.A.M. Frescura and
Basil Hiley Basil J. Hiley (born 1935), is a British quantum physicist and professor emeritus of the University of London. Long-time colleague of David Bohm, Hiley is known for his work with Bohm on implicate orders and for his work on algebraic description ...
have summarized the reasons for the introduction of the principle of complementarity in physics as follows: Complementarity was a central feature of Bohr's reply to the
EPR paradox EPR may refer to: Science and technology * EPR (nuclear reactor), European Pressurised-Water Reactor * EPR paradox (Einstein–Podolsky–Rosen paradox), in physics * Earth potential rise, in electrical engineering * East Pacific Rise, a mid-oce ...
, an attempt by
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
, Boris Podolsky and
Nathan Rosen Nathan Rosen (Hebrew: נתן רוזן; March 22, 1909 – December 18, 1995) was an American-Israeli physicist noted for his study on the structure of the hydrogen atom and his work with Albert Einstein and Boris Podolsky on entangled wave functio ...
to argue that quantum particles must have position and momentum even without being measured and so quantum mechanics must be an incomplete theory. The
thought experiment A thought experiment is a hypothetical situation in which a hypothesis, theory, or principle is laid out for the purpose of thinking through its consequences. History The ancient Greek ''deiknymi'' (), or thought experiment, "was the most anc ...
proposed by Einstein, Podolsky and Rosen involved producing two particles and sending them far apart. The experimenter could choose to measure either the position or the momentum of one particle. Given that result, they could in principle make a precise prediction of what the corresponding measurement on the other, faraway particle would find. To Einstein, Podolsky and Rosen, this implied that the faraway particle must have precise values of both quantities whether or not that particle is measured in any way. Bohr argued in response that the deduction of a position value could not be transferred over to the situation where a momentum value is measured, and vice versa. Later expositions of complementarity by Bohr include a 1938 lecture in
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and a 1949 article written for a
festschrift In academia, a ''Festschrift'' (; plural, ''Festschriften'' ) is a book honoring a respected person, especially an academic, and presented during their lifetime. It generally takes the form of an edited volume, containing contributions from the ...
honoring Albert Einstein. It was also covered in a 1953 essay by Bohr's collaborator Léon Rosenfeld.


Mathematical formalism

Complementarity is mathematically expressed by the operators that represent the observable quantities being measured failing to
commute Commute, commutation or commutative may refer to: * Commuting, the process of travelling between a place of residence and a place of work Mathematics * Commutative property, a property of a mathematical operation whose result is insensitive to th ...
: :\left hat, \hat\right:= \hat\hat - \hat\hat \neq \hat. Observables corresponding to non-commuting operators are called ''incompatible observables''. Incompatible observables cannot have a complete set of common eigenstates. Note that there can be some simultaneous eigenstates of \hat and \hat, but not enough in number to constitute a complete basis. The
canonical commutation relation In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). For example, hat x,\hat p_ ...
:\left hat, \hat\right= i\hbar implies that this applies to position and momentum. Likewise, an analogous relationship holds for any two of the
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally ...
observables defined by the
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used ...
; measurements of spin along perpendicular axes are complementary. This has been generalized to discrete observables with more than two possible outcomes using mutually unbiased bases, which provide complementary observables defined on finite-dimensional
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s.


See also

* Copenhagen interpretation *
Canonical coordinates In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of cl ...
* Interpretations of quantum mechanics * Measurement in quantum mechanics *
Wave–particle duality Wave–particle duality is the concept in quantum mechanics that every particle or quantum entity may be described as either a particle or a wave. It expresses the inability of the classical physics, classical concepts "particle" or "wave" to fu ...


References


Further reading

* Berthold-Georg Englert,
Marlan O. Scully Marlan Orvil Scully (born August 3, 1939) is an American physicist best known for his work in theoretical quantum optics. He is a professor at Texas A&M University and Princeton University. Additionally, in 2012 he developed a lab at the Baylor ...
&
Herbert Walther Herbert Walther (January 19, 1935 in Ludwigshafen/Rhein, Germany – July 22, 2006 in Munich) was a leader in the fields of quantum optics and laser physics. He was a founding director of the Max Planck Institute of Quantum Optics (MPQ) in Garching ...
, ''Quantum Optical Tests of Complementarity'', Nature, Vol 351, pp 111–116 (9 May 1991) and (same authors) ''The Duality in Matter and Light'' Scientific American, pg 56–61, (December 1994). * Niels Bohr, ''Causality and Complementarity: supplementary papers edited by Jan Faye and Henry J. Folse. The Philosophical Writings of Niels Bohr, Volume IV''. Ox Bow Press. 1998. *


External links


Discussions with Einstein on Epistemological Problems in Atomic Physics


{{Authority control Quantum mechanics Niels Bohr Dichotomies Scientific laws