In
condensed matter physics
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the sub ...
, the independent electron approximation is a simplification used in complex systems, consisting of many
electrons
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have no ...
, that approximates the electron-electron interaction in crystals as
null
Null may refer to:
Science, technology, and mathematics Computing
* Null (SQL) (or NULL), a special marker and keyword in SQL indicating that something has no value
* Null character, the zero-valued ASCII character, also designated by , often use ...
. It is a requirement for both the
free electron model
In solid-state physics, the free electron model is a quantum mechanical model for the behaviour of charge carriers in a metallic solid. It was developed in 1927, principally by Arnold Sommerfeld, who combined the classical Drude model with quant ...
and the
nearly-free electron model
In solid-state physics, the nearly free electron model (or NFE model) or quasi-free electron model is a quantum mechanical model of physical properties of electrons that can move almost freely through the crystal lattice of a solid. The model ...
, where it is used alongside
Bloch's theorem
In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who d ...
.
In
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, this approximation is often used to simplify a quantum
many-body problem
The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. ''Microscopic'' here implies that quantum mechanics has to be used to provid ...
into single-particle approximations.
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
Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlike ...
is
BCS theory
BCS theory or Bardeen–Cooper–Schrieffer theory (named after John Bardeen, Leon Cooper, and John Robert Schrieffer) is the first microscopic theory of superconductivity since Heike Kamerlingh Onnes's 1911 discovery. The theory describes sup ...
, in which the attraction of pairs of electrons to each other, termed "
Cooper pair
In condensed matter physics, a Cooper pair or BCS pair (Bardeen–Cooper–Schrieffer pair) is a pair of electrons (or other fermions) bound together at low temperatures in a certain manner first described in 1956 by American physicist Leon Cooper ...
s", is the mechanism behind superconductivity. One major effect of electron-electron interactions is that electrons distribute around the ions so that they
screen
Screen or Screens may refer to:
Arts
* Screen printing (also called ''silkscreening''), a method of printing
* Big screen, a nickname associated with the motion picture industry
* Split screen (filmmaking), a film composition paradigm in which mul ...
the ions in the lattice from other electrons.
Quantum treatment
For an example of the Independent electron approximation's usefulness in
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, consider an ''N''-atom crystal with one free electron per atom (each with
atomic number
The atomic number or nuclear charge number (symbol ''Z'') of a chemical element is the charge number of an atomic nucleus. For ordinary nuclei, this is equal to the proton number (''n''p) or the number of protons found in the nucleus of every ...
''Z''). Neglecting spin, the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
of the system takes the form:
:
, where
is the
reduced Planck's constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...
, e is the
elementary charge
The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundame ...
, ''m
e'' is the
electron rest mass
The electron mass (symbol: ''m''e) is the mass of a stationary electron, also known as the invariant mass of the electron. It is one of the fundamental constants of physics. It has a value of about or about , which has an energy-equivalent of a ...
, and
is the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
operator for electron ''i''. The capitalized
is the ''I''-th
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
location (the equilibrium position of the ''I''-th nuclei) and the lowercase
is the ''i''-th electron position.
The first term in parentheses is called the
kinetic energy operator while the last two are simply the
Coulomb interaction
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventiona ...
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.
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.
This is known as the ''independent electron approximation''.
Bloch's theorem relies on this approximation by setting the effective potential term to a periodic potential of the form
which satisfies
, where
is any
reciprocal lattice
In physics, the reciprocal lattice represents the Fourier transform of another lattice (group) (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is a periodic spatial fu ...
vector (see
Bloch's theorem
In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who d ...
).
This approximation can be formalized using methods from the
Hartree-Fock approximation or
density functional theory
Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body ...
.
See also
*
Strongly correlated material
Strongly correlated materials are a wide class of compounds that include insulators and electronic materials, and show unusual (often technologically useful) electronic and magnetic properties, such as metal-insulator transitions, heavy fermion ...
References
* Omar, M. Ali (1994). Elementary Solid State Physics, 4th ed. Addison Wesley. .
Electron
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