Complementarity (physics)

In physics, complementarity is a conceptual aspect of quantum mechanics that Niels Bohr regarded as an essential feature of the theory. The complementarity principle holds that certain pairs of complementary properties cannot all be observed or measured simultaneously. For example, position and momentum, frequency and lifetime, or optical phase and photon number. In contemporary terms, complementarity encompasses both the uncertainty principle and wave-particle duality. 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, 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.

In physics, complementarity is a conceptual aspect of quantum mechanics that Niels Bohr regarded as an essential feature of the theory. The complementarity principle holds that certain pairs of complementary properties cannot all be observed or measured simultaneously. For example, position and momentum, frequency and lifetime, or optical phase and photon number. In contemporary terms, complementarity encompasses both the uncertainty principle and wave-particle duality. 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, 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== ==Mathematical formalism== For Bohr, complementarity was the "ultimate reason" behind the uncertainty principle. All attempts to grapple with atomic phenomena using classical physics were eventually frustrated, he wrote, leading to the recognition that those phenomena have "complementary aspects". But classical physics can be generalized to address this, and with "astounding simplicity", by describing physical quantities using non-commutative algebra. This mathematical expression of complementarity builds on the work of Hermann Weyl and Julian Schwinger, starting with Hilbert spaces and unitary transformation, leading to the theorems of mutually unbiased bases. In the mathematical formulation of quantum mechanics, physical quantities that classical mechanics had treated as real-valued variables become self-adjoint operators on a Hilbert space. These operators, called "observables", can fail to commute, in which case they are called "incompatible": \left[\hat{A}, \hat{B}\right] := \hat{A}\hat{B} - \hat{B}\hat{A} \neq \hat{0}. Incompatible observables cannot have a complete set of common eigenstates; there can be some simultaneous eigenstates of \hat{A} and \hat{B} , but not enough in number to constitute a complete basis. The canonical commutation relation \left[\hat{x}, \hat{p}\right] = i\hbar implies that this applies to position and momentum. In a Bohrian view, this is a mathematical statement that position and momentum are complementary aspects. Likewise, an analogous relationship holds for any two of the spin observables defined by the Pauli matrices; measurements of spin along perpendicular axes are complementary. The Pauli spin observables are defined for a quantum system described by a two-dimensional Hilbert space; mutually unbiased bases generalize these observables to Hilbert spaces of arbitrary finite dimension. Two bases \{|a_j\rangle\} and \{|b_k\rangle\} for an N -dimensional Hilbert space are mutually unbiased when |\langle a_j|b_k \rangle|^2 = \frac{1}{N}\ \text{for all}\ j, k = 1, ... N-1. Here the basis vector a_1 , for example, has the same overlap with every b_k ; there is equal transition probability between a state in one basis and any state in the other basis. Each basis corresponds to an observable, and the observables for two mutually unbiased bases are complementary to each other. This leads to a definition of 'Principle of Complementarity' as: The concept of complementarity has also been applied to quantum measurements described by positive-operator-valued measures (POVMs). D^2 + V^2\ \le\ 1 The values of D and V can vary between 0 and 1 individually, but any experiment that combines particle and wave detection will limit one or the other, or both. The detailed definition of the two terms vary among applications, but the relation expresses the verified constraint that efforts to detect particle paths will result in less visible wave interference. ==See also== Copenhagen interpretation Canonical coordinates Conjugate variables Interpretations of quantum mechanics Wave–particle duality ==References== ==Further reading== Berthold-Georg Englert, Marlan O. Scully & Herbert Walther, 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 Einstein's Reply to Criticisms

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