Exchange operator

{{main|Exchange interaction}}

{{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).

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| image1 = Particle exchange 2d anticlockwise.gif

| caption1 = Anticlockwise rotation

| image2 = Particle exchange 2d clockwise.gif

| caption2 = Clockwise rotation

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In three or higher dimensions, 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 time reversal operation. Consider two repeated operations of such a particle exchange:

:$\backslash hat\{P\}\backslash left|x\_1,\; x\_2\backslash right\backslash rangle\; =\; \backslash left|x\_2,\; x\_1\backslash right\backslash rangle$

:$\backslash hat\{P\}^2\backslash left|x\_1,\; x\_2\backslash right\backslash rangle\; =\; \backslash hat\{P\}\backslash left|x\_2,\; x\_1\backslash right\backslash rangle\; =\; \backslash left|x\_1,\; x\_2\backslash right\backslash rangle$

Therefore, $\backslash hat\{P\}$ is not only unitary but also an operator square root of 1, which leaves the possibilities

:$\backslash hat\{P\}\backslash left|x\_1,\; x\_2\backslash right\backslash rangle\; =\; \backslash pm\; \backslash left|x\_2,\; x\_1\backslash right\backslash rangle\backslash ,.$

Both signs are realized in nature. Particles satisfying the case of +1 are called bosons, and particles satisfying the case of −1 are called fermions. The spin–statistics theorem dictates that all particles with integer spin are bosons whereas all particles with half-integer spin are fermions.

The exchange operator commutes with the 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 $\backslash hat\{P\}$ 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.

A modified exchange operator is defined in the Hartree–Fock method of quantum chemistry, in order to estimate the exchange energy arising from the exchange statistics described above. In this method, one often defines an energetic exchange operator as:

:$\backslash hat\; K\_j\; (x\_1)\; f\_i(x\_1)=\; \backslash phi\_j(x\_1)\; \backslash int\; \{\; \backslash frac\{\backslash phi\_j^*(x\_2)f\_i(x\_2)\}\{r\_\{12\}\}\backslash mathrm\{d\}x\_2\}$

where $\backslash hat\; K\_j\; (x\_1)$ is the one-electron exchange operator, and $f(x\_1)$,$f(x\_2)$ are the one-electron wavefunctions acted upon by the exchange operator as functions of the electron positions, and $\backslash phi\_j(x\_1)$ and $\backslash phi\_j(x\_2)$ are the one-electron wavefunction of the $j$-th electron as functions of the positions of the electrons. Their separation is denoted $r\_\{12\}$.Levine, I.N., Quantum Chemistry (4th ed., Prentice Hall 1991) p.403. {{ISBN|0-205-12770-3}} 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".

• Exchange interaction
• Hamiltonian (quantum mechanics)
• Coulomb operator
• Exchange symmetry or permutation symmetry

{{Reflist}}

• {{cite journal|title=A new energy decomposition scheme for molecular interactions within the Hartree-Fock approximation|author1=K. Kitaura |author2=K. Morokuma |year=2004|journal=International Journal of Quantum Chemistry|publisher=Wiley|volume=10|issue=2|pages=325–340|doi=10.1002/qua.560100211}}
• {{cite journal|title=Good semiconductor band gaps with a modified local-density approximation|author1=Bylander, D. M. |author2=Kleinman, Leonard |year=1990|journal=Physical Review B |volume=41|issue=11|pages=7868–7871|doi=10.1103/PhysRevB.41.7868|pmid=9993089 |bibcode=1990PhRvB..41.7868B}}
• {{cite journal|title=Exchange Operator Formalism for Integrable Systems of Particles|author=A.P. Polychronakos|year=1992|journal=Phys. Rev. Lett. |volume=69|issue=5 |pages=703–705|doi= 10.1103/PhysRevLett.69.703|pmid=10047011 |arxiv=hep-th/9202057|bibcode=1992PhRvL..69..703P|s2cid=14319416 }}
• {{cite journal|title=On a new exchange potential|year=1957|journal=Acta Physica Academiae Scientiarum Hungaricae|volume=7|issue=3|pages=357–364|doi=10.1007/BF03156345 |last1=Szépfalusy |first1=P. |s2cid=124672448 }}
• {{cite journal|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|pages=87–103|bibcode=1958AnPhy...4...87N|doi=10.1016/0003-4916(58)90039-3}}
• {{cite web|title=The Hartree-Fock Equation|url=http://vergil.chemistry.gatech.edu/notes/hf-intro/node7.html}}

• [http://www.cmth.ph.ic.ac.uk/people/p.haynes/thesis/node14.html 2.3.Identical particles, P. Haynes] {{Webarchive|url=https://web.archive.org/web/20130701222719/http://www.cmth.ph.ic.ac.uk/people/p.haynes/thesis/node14.html |date=2013-07-01 }}
• [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]

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Category:Quantum chemistry