Exchange interaction

From Wikipedia, the free encyclopedia

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 appearing "closer together" than would be expected of distinguishable particles, and in identical particles with spatially antisymmetric wave functions appearing "farther apart".

Although one might naively expect such an interaction to result from 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, that 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 Heisenberg[2] and 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]

Contents

[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 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 antisymmetrising 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 symmetrising 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, VI(r1,r2), is symmetric, and restrict our attention to the vector space spanned by ΨA and ΨS, 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] 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.[8]

[edit] See also

[edit] References

  1. ^ David J. Griffifths, "Introduction to Quantum Mechanics", Second Edition, pp. 207-210
  2. ^ Mehrkörperproblem und Resonanz in der Quantenmechanik, W. Heisenberg, Zeitschrift für Physik 38, #6–7 (June 1926), pp. 411–426. DOI 10.1007/BF01397160.
  3. ^ 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.
  4. ^ Exchange Forces, HyperPhysics, Georgia State University, accessed June 2, 2007.
  5. ^ Derivation of the Heisenberg Hamiltonian, Rebecca Hihinashvili, accessed on line October 2, 2007.
  6. ^ 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.
  7. ^ The Theory of Electric and Magnetic Susceptibilities, J. H. van Vleck, London: Oxford University Press, 1932, chapter XII, section 76.
  8. ^ Exchange interaction, F. Duncan and M. Haldane, AccessScience@McGraw-Hill, DOI 10.1036/1097-8542.247650, dated 2000-IV-10.

[edit] External links