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In physics, the
exchange interaction is a quantum mechanical effect which increases or decreases the
energy of two or more electrons when their
wavefunctions overlap. Arising from the
Pauli exclusion principle, this energy change is the result of the
identical particles,
exchange symmetry, and the
electrostatic force. Exchange interaction effects were discovered independently by Werner HeisenbergMehrkörperproblem und Resonanz in der Quantenmechanik, W. Heisenberg,
Zeitschrift für Physik 38, #6–7 (June 1926), pp. 411–426. DOI 10.1007/BF01397160. and P. A. M. Dirac 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. in
1926.
The exchange interaction is also called the
exchange forcepp. 87–88,
Driving Force: the natural magic of magnets, James D. Livingston, Harvard University Press, 1996. ISBN 0674216458., but is not the same as 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. Exchange Forces, HyperPhysics,
Georgia State University, accessed
June 2,
2007.
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 (physics) 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 \Psi_1(r_1) for the first electron and \Psi_2(r_2) for the second electron. We assume that \Psi_1 and \Psi_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 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, V_I(r_1, r_2), is symmetric, and restrict our attention to the vector space spanned by \Psi_A and \Psi_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 (quantum mechanics), where
S1 and
S2 are the spin (physics) operators of the two electrons. This term, often referred to as the
Heisenberg Hamiltonian, gives one form of the exchange interaction. 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. 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.Exchange interaction, F. Duncan and M. Haldane, AccessScience@McGraw-Hill, DOI 10.1036/1097-8542.247650, dated 2000-IV-10.
See also
References
External links
- Exchange Interaction (PDF)
- Exchange Interaction and Energy
- Exchange Interaction and Exchange Anisotropy
In physics, the
exchange interaction is a quantum mechanical effect which increases or decreases the energy of two or more
electrons when their wavefunctions overlap. Arising from the Pauli exclusion principle, this energy change is the result of the identical particles,
exchange symmetry, and the electrostatic force. Exchange interaction effects were discovered independently by Werner HeisenbergMehrkörperproblem und Resonanz in der Quantenmechanik, W. Heisenberg,
Zeitschrift für Physik 38, #6–7 (June 1926), pp. 411–426. DOI 10.1007/BF01397160. and
P. A. M. Dirac 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. in 1926.
The exchange interaction is also called the
exchange forcepp. 87–88,
Driving Force: the natural magic of magnets, James D. Livingston, Harvard University Press, 1996. ISBN 0674216458., but is not the same as 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. Exchange Forces, HyperPhysics, Georgia State University, accessed
June 2, 2007.
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 (physics) 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 \Psi_1(r_1) for the first electron and \Psi_2(r_2) for the second electron. We assume that \Psi_1 and \Psi_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 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, V_I(r_1, r_2), is symmetric, and restrict our attention to the vector space spanned by \Psi_A and \Psi_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 (quantum mechanics), where
S1 and
S2 are the spin (physics) operators of the two electrons. This term, often referred to as the
Heisenberg Hamiltonian, gives one form of the exchange interaction. 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. 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.Exchange interaction, F. Duncan and M. Haldane, AccessScience@McGraw-Hill, DOI 10.1036/1097-8542.247650, dated 2000-IV-10.
See also
References
External links
- Exchange Interaction (PDF)
- Exchange Interaction and Energy
- Exchange Interaction and Exchange Anisotropy
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