Breit equation
Encyclopedia
The Breit equation is a relativistic
wave equation
derived by Gregory Breit
in 1929 based on the Dirac equation
, which formally describes two or more massive spin
-1/2 particles (electron
s, for example) interacting electromagnetically to the first order in perturbation theory
. It accounts for magnetic interactions and retardation effects to the order of 1/c2. When other quantum electrodynamic effects are negligible, this equation has been shown to give results in good agreement with experiment. It was originally derived from the Darwin Lagrangian but later vindicated by the Wheeler-Feynman absorber theory and eventually quantum electrodynamics
.
, but also in terms of relativity theory as it is not completely invariant with respect to the Lorentz transformation
. Just as does the Dirac equation
, it treats nuclei as point sources of an external field for the particles it describes. For N particles, the Breit equation has the form (rij is the distance between particle i and j):
,
where
is the Dirac hamiltonian (see Dirac equation
) for particle i at position ri and φ(ri) is the scalar potential at that position; qi is the charge of the particle, thus for electrons qi = - e.
The one-electron Dirac hamiltonians of the particles, along with their instantaneous Coulomb interactions 1/rij, form the Dirac-Coulomb operator. To this, Breit added the operator (now known as the (frequency-independent) Breit operator):
,
where the Dirac matrices for electron i: a(i) = [αx(i),αy(i),αz(i)]. The two terms in the Breit operator account for retardation effects to the first order.
The wave function Ψ in the Breit equation is a spinor
with 4N elements, since each electron is described by a Dirac bispinor
with 4 elements as in the Dirac equation
and total wave function is the cartesian product of these.
):
,
in which the consecutive partial operators are:
where:
and
Theory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....
wave equation
Relativistic wave equations
Before the creation of quantum field theory, physicists attempted to formulate versions of the Schrödinger equation which were compatible with special relativity...
derived by Gregory Breit
Gregory Breit
Gregory Breit was a Russian-born American physicist and professor at universities in New York, Wisconsin, Yale, and Buffalo...
in 1929 based on the Dirac equation
Dirac equation
The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...
, which formally describes two or more massive spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
-1/2 particles (electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s, for example) interacting electromagnetically to the first order in perturbation theory
Perturbation theory
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
. It accounts for magnetic interactions and retardation effects to the order of 1/c2. When other quantum electrodynamic effects are negligible, this equation has been shown to give results in good agreement with experiment. It was originally derived from the Darwin Lagrangian but later vindicated by the Wheeler-Feynman absorber theory and eventually quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...
.
Introduction
The Breit equation is not only an approximation in terms of quantum mechanicsQuantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, but also in terms of relativity theory as it is not completely invariant with respect to the Lorentz transformation
Lorentz transformation
In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...
. Just as does the Dirac equation
Dirac equation
The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...
, it treats nuclei as point sources of an external field for the particles it describes. For N particles, the Breit equation has the form (rij is the distance between particle i and j):
,where
is the Dirac hamiltonian (see Dirac equation
Dirac equation
The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...
) for particle i at position ri and φ(ri) is the scalar potential at that position; qi is the charge of the particle, thus for electrons qi = - e.
The one-electron Dirac hamiltonians of the particles, along with their instantaneous Coulomb interactions 1/rij, form the Dirac-Coulomb operator. To this, Breit added the operator (now known as the (frequency-independent) Breit operator):
,where the Dirac matrices for electron i: a(i) = [αx(i),αy(i),αz(i)]. The two terms in the Breit operator account for retardation effects to the first order.
The wave function Ψ in the Breit equation is a spinor
Spinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
with 4N elements, since each electron is described by a Dirac bispinor
Bispinor
In physics, bispinor is a four-component object which transforms under the ⊕ representation of the covariance group of special relativity...
with 4 elements as in the Dirac equation
Dirac equation
The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...
and total wave function is the cartesian product of these.
Breit hamiltonians
The total hamiltonian of the Breit equation, sometimes called the Dirac-Coulomb-Breit hamiltonian (HDCB) can be decomposed into the following practical energy operators for electrons in electric and magnetic fields (also called the Breit-Pauli hamiltonian) , which have well-defined meanings in the interaction of molecules with magnetic fields (for instance for nuclear magnetic resonanceNuclear magnetic resonance
Nuclear magnetic resonance is a physical phenomenon in which magnetic nuclei in a magnetic field absorb and re-emit electromagnetic radiation...
):
,in which the consecutive partial operators are:
-
is the nonrelativistic hamiltonian (
is the stationary mass of particle i). -
is connected to the dependence of mass on velocity: 
. -
is a correction that partly accounts for retardation and can be described as the interaction between the magnetic dipole moments of the particles, which arise from the orbital motion of charges (also called orbit-orbit interaction). -
is the classical interaction between the orbital magnetic moments (from the orbital motion of charge) and spin magnetic moments (also called spin-orbit interactionSpin-orbit interactionIn quantum physics, the spin-orbit interaction is any interaction of a particle's spin with its motion. The first and best known example of this is that spin-orbit interaction causes shifts in an electron's atomic energy levels due to electromagnetic interaction between the electron's spin and...
). The first term describes the interaction of a particles spin with its own orbital moment (F(ri) is the electric field at the particle's position), and the second term between two different particles. -
is a nonclassical term characteristic for Dirac theory, sometimes called the DarwinCharles Galton DarwinSir Charles Galton Darwin, KBE, MC, FRS was an English physicist, the grandson of Charles Darwin. He served as director of the National Physical Laboratory during the Second World War.-Early life:...
term. -
is the magnetic moment spin-spin interaction. The first term is called the contact interactionFermi contact interactionThe Fermi contact interaction is the magnetic interaction between an electron and an atomic nucleus when the electron is inside that nucleus. It is of magnitude...
, because it is nonzero only when the particles are at the same position; the second term is the interaction of the classical dipole-dipole type. -
is the interaction between spin and orbital magnetic moments with an external magnetic field H.
where:
and
See also
- Bethe-Salpeter equationBethe-Salpeter equationThe Bethe–Salpeter equation, named after Hans Bethe and Edwin Salpeter, describes the bound states of a two-body quantum field theoretical system in a relativistically covariant formalism...
- Darwin Lagrangian
- Wheeler-Feynman absorber theory