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, 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.
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.
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 for the first electron and for the second electron. We assume that and 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:
When J is positive, the exchange energy favors electrons with parallel spins; this is a primary cause of ferromagnetism in materials such as iron. In fact, when the interaction VI is purely due to Coulomb repulsion of electrons (i.e. ), J is always positive (unless the wavefunctions do not overlap at all, in which case J is zero).
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.
Entangling identical bosons in optical tweezers via exchange interaction.(Quantum Information Processing: Article)(Report)
Apr 01, 2008; Abstract: We first devise a scheme to perform a universal entangling gate via controlled collisions between pairs of atomic...