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	~~Andera~~ is what ~~you can contact her~~ but ~~she~~ [~~http~~:/~~/alles~~-~~herunterladen~~.de/~~excellent~~-~~advice~~-~~for~~-~~picking~~-the-~~ideal~~-~~hobby~~/ ~~psychic solutions by lynne~~] ~~no means really favored that~~ title. ~~Office supervising is what she does~~ for a ~~living~~. ~~Kentucky is where I~~'~~ve always been residing~~. ~~Playing badminton is a factor that he is completely addicted to~~.		{{Unreferenced\|date=December 2009}}
			{{quantum mechanics}}

			In [[quantum mechanics]], the '''exchange operator''' <math>\hat{P}</math> is a [[operator (quantum mechanics)\|quantum mechanical operator]] that acts on states in [[Fock space]]. The exchange operator acts by switching the labels on any two [[identical particles]] described by the joint position [[quantum state]] <math>\left\|x_1, x_2\right\rangle</math>. Since the particles are identical, the notion of [[exchange symmetry]] requires that the exchange operator be [[unitarity_(physics)\|unitary]].

			==Construction==

			{{multiple image
			\| footer = Exchange of two particles in 2 + 1 spacetime by rotation. The rotations are inequivalent, since one cannot be deformed into the other (without the worldlines leaving the plane, an impossibility in 2d space).
			\| width = 150

			\| image1 = Particle exchange 2d anticlockwise.gif
			\| caption1 = Anticlockwise rotation

			\| image2 = Particle exchange 2d clockwise.gif
			\| caption2 = Clockwise rotation
			}}

			In three or higher [[dimension]]s, the exchange operator can represent a literal exchange of the positions of the pair of particles by motion of the particles in an [[adiabatic process]], with all other particles held fixed. Such motion is often not carried out in practice. Rather, the operation is treated as a "what if" similar to a [[parity inversion]] or [[T-symmetry\|time reversal]] operation. Consider two repeated operations of such a particle exchange:

			:<math>\hat{P}^2\left\|x_1, x_2\right\rangle = \hat{P}\left\|x_2, x_1\right\rangle = \left\|x_1, x_2\right\rangle</math>

			Therefore <math>\hat{P}</math> is not only unitary but also an [[square root of a matrix\|operator square root]] of 1, which leaves the possibilities

			:<math>\hat{P}\left\|x_1, x_2\right\rangle = \pm \left\|x_1, x_2\right\rangle\,.</math>

			Both signs are realized in nature. Particles satisfying the case of +1 are called ''[[boson]]s'', and particles satisfying the case of −1 are called ''[[fermion]]s''. The [[spin–statistics theorem]] dictates that all particles with integer [[spin (physics)\|spin]] are bosons whereas all particles with half-integer spin are fermions.

			The exchange operator commutes with the [[Hamiltonian (quantum mechanics)\|Hamiltonian]] and is therefore a [[conserved quantity]]. Therefore it is always possible and usually most convenient to choose a basis in which the states are eigenstates of the exchange operator. Such a state is either completely symmetric under exchange of all identical bosons or completely antisymmetric under exchange of all identical fermions of the system. To do so for fermions, for example, the [[antisymmetrizer]] builds such a completely antisymmetric state.

			In 2 dimensions, the adiabatic exchange of particles is not necessarily possible. Instead, the eigenvalues of the exchange operator may be complex phase factors (in which case <math>\hat{P}</math> is not Hermitian), see [[anyon]] for this case. The exchange operator is not well defined in a strictly 1-dimensional system, though there are constructions of 1-dimensional networks that behave as effective 2-dimensional systems.

			==Quantum chemistry==
			In the context of [[quantum chemistry]], a common problem is to estimate modifications to the Hamiltonian arising from the exchange statistics described above. Specifically, in the [[Hartree–Fock method]], one often defines an energetic exchange operator as:

			:<math>\hat K_j (x_1) f(x_1)= \phi_j(x_1) \int { \frac{\phi_j^*(x_2)f(x_2)}{r_{12}}dx_2}</math>

			where <math>\hat K_j (x_1)</math> is the one-electron exchange operator, and <math> f(x_1) </math>,<math> f(x_2) </math> are the one-electron [[wavefunction]]s acted upon by the exchange operator as functions of the electron positions, and <math> \phi_j(x_1) </math> and <math> \phi_j(x_2) </math> are the one-electron wavefunction of the ''j''th electron as functions of the positions of the electrons. Their separation is denoted ''r''<sub>12</sub>. The labels 1 and 2 are only for a notational convenience, since physically there is no way to keep track of "which electron is which".

			==See also==

			*[[Hamiltonian (quantum mechanics)]]
			*[[Coulomb operator]]

			==References==

			*{{cite article\|title=A new energy decomposition scheme for molecular interactions within the Hartree-Fock approximation\|author= K. Kitaura, K. Morokuma\|year=2004\|journal=International journal of quantum chemistry\|publisher=Wiley\|
			volume=10\|issue=2\|page=325–340\|url=http://onlinelibrary.wiley.com/doi/10.1002/qua.560100211/abstract;jsessionid=ACF30D173A2ECCB52C43518A1E2CC834.d03t02?deniedAccessCustomisedMessage=&userIsAuthenticated=false\|doi=10.1002/qua.560100211\|arxiv=}}

			*{{cite article\|title=Good semiconductor band gaps with a modified local-density approximation\|author= Bylander, D. M.; Kleinman, Leonard\|year=1990\|journal=Physical Review B (Condensed Matter)\|publisher=\|
			volume=41\|issue=11\|page=7868-7871\|url=http://adsabs.harvard.edu/abs/1990PhRvB..41.7868B\|doi=10.1103/PhysRevB.41.7868\|arxiv=}}

			*{{cite article\|title=Exchange Operator Formalism for Integrable Systems of Particles\|author=A.P. Polychronakos\|year=1992\|journal=Phys.Rev.Lett\|publisher=\|
			volume=69\|issue=\|page=703-705\|url=http://arxiv.org/pdf/hep-th/9202057v1.pdf\|doi= 10.1103/PhysRevLett.69.703\|arxiv=hep-th/9202057}}

			*{{cite book\|title=A modern approach to quantum mechanics
			\|author=J.S. Townsend\|edition=2\|series=International series in pure and applied physics\|year=2000\|journal=\|publisher=University Science Books\|volume=69\|issue=\|page=342\|url=http://books.google.co.uk/books?id=3_7uriPX028C&pg=PA342&lpg=PA342&dq=Exchange+operator+quantum+mechanics&source=bl&ots=Tj8ZR40dhi&sig=Ykqr2hGm3udQPB7I4s88lNdxnkI&hl=en&sa=X&ei=XYmeUf7ZJ8S0PNu2gcgG&ved=0CDoQ6AEwAQ#v=onepage&q=Exchange%20operator%20quantum%20mechanics&f=false\|isbn=1891389130}}

			*{{cite article\|title=On a new exchange potential\|author=\|year=1957\|publisher=Acta Physica Academiae Scientiarum Hungaricae\|
			volume=7\|issue=3\|page=357-364\|url=http://link.springer.com/article/10.1007%2FBF03156345}}

			*{{cite article\|title=The Heisenberg exchange operator for ferromagnetic and antiferromagnetic systems\|author=R.K Nesbet\|year=1958\|journal=Annals of Physics\|location=Lincoln, Massachusetts, USA\|publisher=Elsevier\|
			volume=4\|issue=1\|page=87–103\|url=http://www.sciencedirect.com/science/article/pii/0003491658900393}}

			*{{cite website\|title=The Hartree-Fock Equation\|url=http://vergil.chemistry.gatech.edu/notes/hf-intro/node7.html}}

			==External links==

			*[http://www.cmth.ph.ic.ac.uk/people/p.haynes/thesis/node14.html 2.3.''Identical particles'', P. Haynes]
			*[http://www.sonic.net/~rknop/php/astronomy/enma/enma_v0.29_12.pdf Chapter 12, ''Multiple Particle States'']
			*[http://www.av8n.com/physics/exchange.htm ''Exchange of Identical and Possibly Indistinguishable Particles'', J. Denker]

			{{DEFAULTSORT:Exchange Operator}}
			[[Category:Quantum chemistry]]

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