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In physics, the exchange interaction is a quantum mechanical effect which increases or decreases the expectation value of the energy or distance between two or more identical particles when their wave functions overlap. For example, the exchange interaction results in identical particles with spatially symmetric wave functions (bosons) appearing "closer together" than would be expected of distinguishable particles, and in identical particles with spatially antisymmetric wave functions (fermions) appearing "farther apart". It is the mechanism that is responsible for ferromagnetism, among other consequences.

Although sometimes erroneously described as a force, the exchange interaction is a purely quantum mechanical effect without any analog in classical mechanics. It is the result of the fact that the wave function of indistinguishable particles is subject to exchange symmetry, that is, the wave function describing two particles that cannot be distinguished must be either unchanged (symmetric) or inverted in sign (antisymmetric) if the labels of the two particles are changed.

For example, if the expectation value of the distance between two particles in a spatially symmetric or antisymmetric state is calculated, the exchange interaction may be seen.^[1]

Both bosons and fermions can experience the exchange interaction provided that the particles in question are indistinguishable.

Exchange interaction effects were discovered independently by physicists Werner Heisenberg^[2] and P. A. M. Dirac^[3] in 1926.

The exchange interaction is sometimes called the exchange force, but it is not a true force and should not be confused with the exchange forces produced by the exchange of force carriers, such as the electromagnetic force produced between two electrons by the exchange of a photon, or the strong force between two quarks produced by the exchange of a gluon.^[4]

[edit] Overview

Quantum mechanical particles are classified as bosons or fermions. 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. Multiple bosons may occupy the same quantum state; by the Pauli exclusion principle, however, no two fermions can occupy the same state. 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 using an antisymmetric combination of the product 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 using a symmetric combination of the product 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 term, often referred to as the Heisenberg Hamiltonian, gives one form of the exchange interaction.^[5]^[6]^[7]

When J is positive, the exchange energy favors electrons with parallel spins; this is a primary cause of ferromagnetism in materials such as iron.^[8] In fact, when the interaction V_I is purely due to Coulomb repulsion of electrons (i.e. $V I (r 1, r 2) = e 2 / | r 1 - r 2 |$ ), J is always positive (unless the wavefunctions do not overlap at all, in which case J is zero).^[9]

When J is negative, the interaction favors electrons with antiparallel spins, potentially causing antiferromagnetism.

Although these consequences of the exchange interaction are magnetic in nature, the cause is not; it is due primarily to electric repulsion and the Pauli exclusion principle. Indeed, in general, the direct magnetic interaction between a pair of electrons (due to their electron magnetic moments) is negligibly small compared to this electric interaction.

Exchange energy splittings are very elusive to calculate for molecular systems at large internuclear distances. However, analytical formulae have been worked out for the hydrogen molecular ion (see references herein).

[edit] See also

[edit] References

^ David J. Griffifths, "Introduction to Quantum Mechanics", Second Edition, pp. 207-210
^ 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.
^ Exchange Forces, HyperPhysics, Georgia State University, accessed June 2, 2007.
^ Derivation of the Heisenberg Hamiltonian, Rebecca Hihinashvili, accessed on line October 2, 2007.
^ 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.
^ Slater, Quantum Theory of Electronic Structure, 1960, Appendix 19.

[edit] External links

[0] David J. Griffifths, "Introduction to Quantum Mechanics", Second Edition, pp. 207-210

[1] Mehrkörperproblem und Resonanz in der Quantenmechanik, W. Heisenberg, Zeitschrift für Physik 38, #6–7 (June 1926), pp. 411–426. DOI 10.1007/BF01397160.

[2] 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.

[3] Exchange Forces, HyperPhysics, Georgia State University, accessed June 2, 2007.

[4] Derivation of the Heisenberg Hamiltonian, Rebecca Hihinashvili, accessed on line October 2, 2007.

[5] 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.

[6] The Theory of Electric and Magnetic Susceptibilities, J. H. van Vleck, London: Oxford University Press, 1932, chapter XII, section 76.

[7] Exchange interaction, F. Duncan and M. Haldane, AccessScience@McGraw-Hill, DOI 10.1036/1097-8542.247650, dated 2000-IV-10.

[8] Slater, Quantum Theory of Electronic Structure, 1960, Appendix 19.

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Contents

[edit] Overview

[edit] See also

[edit] References

[edit] External links