In physics, the exchange interaction is a quantum mechanical effect which increases or decreases the energy of two or more electrons when their wave functions overlap. This energy change is the result of an effect due to the identity of particles, exchange symmetry, and the electrostatic force. Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the fundamental laws of the universe and their precise formulation in a mathematical framework. ... Fig. ... e- redirects here. ... This article discusses the concept of a wavefunction as it relates to quantum mechanics. ... Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. ... Exchange symmetry is derived from a fundamental postulate of quantum statistics, which states that no observable physical quantity should change after exchanging two identical particles. ... Electrostatics is the branch of physics that deals with the force exerted by a static (i. ...

Overview

According to quantum mechanics in three dimensions, every particle must behave as a boson or a fermion. In the former case, two (or more) particles can occupy the same quantum state; in the latter case, the Pauli exclusion principle means that no two particles can occupy the same state. The spin-statistics theorem of quantum field theory demands that all particles with half-integer spin behave as fermions and all particles with integer spin behave as bosons. Thus, since electrons have spin 1/2, they are fermions. This means that the overall wavefunction of a system must be antisymmetric when two electrons are exchanged. In particle physics, bosons, named after Satyendra Nath Bose, are particles having integer spin. ... In particle physics, fermions are particles with half-integer spin. ... A quantum state is any possible state in which a quantum mechanical system can be. ... The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. ... The spin-statistics theorem in quantum mechanics relates the spin of a particle to the statistics obeyed by that particle. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... In mathematics, a half-integer is a number of the form , where is an integer. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ... The integers are commonly denoted by the above symbol. ... e- redirects here. ... This article discusses the concept of a wavefunction as it relates to quantum mechanics. ...

Taking a system with two electrons, we may attempt to model the state of each electron by first assuming the electrons behave independently, and taking wavefunctions in position space of Ψ₁(r₁) for the first electron and Ψ₂(r₂) for the second electron. We assume that Ψ₁ and Ψ₂ are orthogonal, and that each corresponds to an energy eigenstate of its electron. Now, if the overall system has spin 1, the spin wave function is symmetric, and we may construct a wavefunction for the overall system in position space by antisymmetrizing the product of these wavefunctions in position space: This article discusses the concept of a wavefunction as it relates to quantum mechanics. ...

On the other hand, if the overall system has spin 0, the spin wave function is antisymmetric, and we may therefore construct the overall position-space wavefunction by symmetrizing the product of the wavefunctions in position space:

If we assume that the interaction energy between the two electrons, V_I(r₁,r₂), is symmetric, and restrict our attention to the vector space spanned by Ψ_A and Ψ_S, then each of these wavefunctions will yield eigenstates for the system energy, and the difference between their energies will be

Taking into account the different joint spins of these eigenstates, we may model this difference by adding a spin-spin interaction term

to the Hamiltonian, where S₁ and S₂ are the spin operators of the two electrons. This is one form of the exchange interaction.^[1],[2] Despite its form, it is not magnetic in nature. In materials such as iron, this effect favors electrons with parallel spins and is thus a cause of ferromagnetism.^[3] The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ... General Name, Symbol, Number iron, Fe, 26 Chemical series transition metals Group, Period, Block 8, 4, d Appearance lustrous metallic with a grayish tinge Atomic mass 55. ... Ferromagnetism is the phenomenon by which materials, such as iron, in an external magnetic field become magnetized and remain magnetized for a period after the material is no longer in the field. ...

History

Exchange effects were discovered independently by Heisenberg^[4] and Dirac^[5] in 1926. Werner Karl Heisenberg (December 5, 1901 – February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics, and acknowledged to be one of the most important physicists of the twentieth century. ... Paul Adrien Maurice Dirac, (August 8, 1902 - October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... Year 1926 (MCMXXVI) was a common year starting on Friday (link will display the full calendar). ...

See also

• Exchange symmetry
• Pauli exclusion principle

Exchange symmetry is derived from a fundamental postulate of quantum statistics, which states that no observable physical quantity should change after exchanging two identical particles. ... The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. ...

References

1. ^ Quantum Theory of Magnetism: Magnetic Properties of Materials, Robert M. White, 3rd rev. ed., Berlin: Springer-Verlag, 2007, section 2.2.7. ISBN 3-540-65116-0.
2. ^ The Theory of Electric and Magnetic Susceptibilities, J. H. van Vleck, London: Oxford University Press, 1932, chapter XII, section 76.
3. ^ Exchange interaction, F. Duncan and M. Haldane, AccessScience@McGraw-Hill, DOI 10.1036/1097-8542.247650, dated 2000-IV-10.
4. ^ Mehrkörperproblem und Resonanz in der Quantenmechanik, W. Heisenberg, Zeitschrift für Physik 38, #6–7 (June 1926), pp. 411–426. DOI 10.1007/BF01397160.
5. ^ On the Theory of Quantum Mechanics, P. A. M. Dirac, Proceedings of the Royal Society of London, Series A 112, #762 (October 1, 1926), pp. 661—677.

October 1 is the 274th day of the year (275th in leap years) in the Gregorian calendar. ... Year 1926 (MCMXXVI) was a common year starting on Friday (link will display the full calendar). ...

External links

• Exchange Interaction (PDF)
• Exchange Interaction and Energy
• Exchange Interaction and Exchange Anisotropy