Exchange interaction

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This article is about an interaction arising from the Pauli exclusion principle. For interactions arising from exchange of force carriers, see force carrier.

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.

1 Overview
2 History
3 See also
4 References
5 External links

[edit] 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.

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 $Ψ 1 (r 1)$ for the first electron and $Ψ 2 (r 2)$ for the second electron. We assume that $Ψ 1$ and $Ψ 2$ 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:

$\Psi_A(r_1,r_2)=(\Psi_1(r_1) \Psi_2(r_2) - \Psi_2(r_1) \Psi_1(r_2))/\sqrt{2}.$

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:

$\Psi_S(r_1,r_2)=(\Psi_1(r_1) \Psi_2(r_2) + \Psi_2(r_1) \Psi_1(r_2))/\sqrt{2}.$

If we assume that the interaction energy between the two electrons, $V I (r 1, r 2)$ , 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

$J=2\int \Psi_1^{*}(r_1) \Psi_2^{*}(r_2) V_I(r_1, r_2) \Psi_2(r_1) \Psi_1(r_2) \, dr_1\, dr_2.$

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

$-J S_1 \cdot S_2$

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]

[edit] History

Exchange effects were discovered independently by Heisenberg^[4] and Dirac^[5] in 1926.

[edit] See also

[edit] References

^ 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.
^ The Theory of Electric and Magnetic Susceptibilities, J. H. van Vleck, London: Oxford University Press, 1932, chapter XII, section 76.
^ Exchange interaction, F. Duncan and M. Haldane, AccessScience@McGraw-Hill, DOI 10.1036/1097-8542.247650, dated 2000-IV-10.
^ Mehrkörperproblem und Resonanz in der Quantenmechanik, W. Heisenberg, Zeitschrift für Physik 38, #6–7 (June 1926), pp. 411–426. DOI 10.1007/BF01397160.
^ 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.