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Independent electron approximation

In condensed matter physics, the independent electron approximation is a simplification used in complex systems, consisting of many electrons, that approximates the electron–electron interaction in crystals as null. It is a requirement for both the free electron model and the nearly-free electron model, where it is used alongside Bloch's theorem.[1] In quantum mechanics, this approximation is often used to simplify a quantum many-body problem into single-particle approximations.[1]

While this simplification holds for many systems, electron–electron interactions may be very important for certain properties in materials. For example, the theory covering much of superconductivity is BCS theory, in which the attraction of pairs of electrons to each other, termed "Cooper pairs", is the mechanism behind superconductivity. One major effect of electron–electron interactions is that electrons distribute around the ions so that they screen the ions in the lattice from other electrons.[citation needed]

Quantum treatment edit

For an example of the Independent electron approximation's usefulness in quantum mechanics, consider an N-atom crystal with one free electron per atom (each with atomic number Z). Neglecting spin, the Hamiltonian of the system takes the form:[1]

 

where   is the reduced Planck constant, e is the elementary charge, me is the electron rest mass, and   is the gradient operator for electron i. The capitalized   is the Ith lattice location (the equilibrium position of the Ith nuclei) and the lowercase   is the ith electron position.

The first term in parentheses is called the kinetic energy operator while the last two are simply the Coulomb interaction terms for electron–nucleus and electron–electron interactions, respectively. If the electron–electron term were negligible, the Hamiltonian could be decomposed into a set of N decoupled Hamiltonians (one for each electron), which greatly simplifies analysis. The electron–electron interaction term, however, prevents this decomposition by ensuring that the Hamiltonian for each electron will include terms for the position of every other electron in the system.[1] If the electron–electron interaction term is sufficiently small, however, the Coulomb interactions terms can be approximated by an effective potential term, which neglects electron–electron interactions.[1] This is known as the independent electron approximation.[1] Bloch's theorem relies on this approximation by setting the effective potential term to a periodic potential of the form   that satisfies  , where   is any reciprocal lattice vector (see Bloch's theorem).[1] This approximation can be formalized using methods from the Hartree–Fock approximation or density functional theory.[1]

See also edit

References edit

  1. ^ a b c d e f g h Girvin, Steven M.; Yang, Kun (2019). Modern Condensed Matter Physics (1 ed.). Cambridge University Press. pp. 105–117. ISBN 978-1-107-13739-4.

independent, electron, approximation, condensed, matter, physics, independent, electron, approximation, simplification, used, complex, systems, consisting, many, electrons, that, approximates, electron, electron, interaction, crystals, null, requirement, both,. In condensed matter physics the independent electron approximation is a simplification used in complex systems consisting of many electrons that approximates the electron electron interaction in crystals as null It is a requirement for both the free electron model and the nearly free electron model where it is used alongside Bloch s theorem 1 In quantum mechanics this approximation is often used to simplify a quantum many body problem into single particle approximations 1 While this simplification holds for many systems electron electron interactions may be very important for certain properties in materials For example the theory covering much of superconductivity is BCS theory in which the attraction of pairs of electrons to each other termed Cooper pairs is the mechanism behind superconductivity One major effect of electron electron interactions is that electrons distribute around the ions so that they screen the ions in the lattice from other electrons citation needed Quantum treatment editFor an example of the Independent electron approximation s usefulness in quantum mechanics consider an N atom crystal with one free electron per atom each with atomic number Z Neglecting spin the Hamiltonian of the system takes the form 1 H i 1 N ℏ 2 i 2 2 m e I 0 N e 2 Z r i R I 1 2 i j N e 2 r i r j displaystyle mathcal H sum i 1 N left frac hbar 2 nabla i 2 2m e sum I 0 N frac e 2 Z left mathbf r i mathbf R I right frac 1 2 sum i neq j N frac e 2 left mathbf r i mathbf r j right right nbsp where ℏ displaystyle hbar nbsp is the reduced Planck constant e is the elementary charge me is the electron rest mass and i displaystyle nabla i nbsp is the gradient operator for electron i The capitalized R I displaystyle mathbf R I nbsp is the Ith lattice location the equilibrium position of the Ith nuclei and the lowercase r i displaystyle mathbf r i nbsp is the ith electron position The first term in parentheses is called the kinetic energy operator while the last two are simply the Coulomb interaction terms for electron nucleus and electron electron interactions respectively If the electron electron term were negligible the Hamiltonian could be decomposed into a set of N decoupled Hamiltonians one for each electron which greatly simplifies analysis The electron electron interaction term however prevents this decomposition by ensuring that the Hamiltonian for each electron will include terms for the position of every other electron in the system 1 If the electron electron interaction term is sufficiently small however the Coulomb interactions terms can be approximated by an effective potential term which neglects electron electron interactions 1 This is known as the independent electron approximation 1 Bloch s theorem relies on this approximation by setting the effective potential term to a periodic potential of the form V r displaystyle V mathbf r nbsp that satisfies V r R j V r displaystyle V mathbf r mathbf R j V mathbf r nbsp where R j displaystyle mathbf R j nbsp is any reciprocal lattice vector see Bloch s theorem 1 This approximation can be formalized using methods from the Hartree Fock approximation or density functional theory 1 See also editStrongly correlated materialReferences edit a b c d e f g h Girvin Steven M Yang Kun 2019 Modern Condensed Matter Physics 1 ed Cambridge University Press pp 105 117 ISBN 978 1 107 13739 4 Omar M Ali 1994 Elementary Solid State Physics 4th ed Addison Wesley ISBN 978 0 201 60733 8 Retrieved from https en wikipedia org w index php title Independent electron approximation amp oldid 1220431814, wikipedia, wiki, book, books, library,

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