# Breit equation

﻿
Breit equation

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 (electrons, 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.

Introduction

The Breit equation is not only an approximation in terms of quantum mechanics, 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"):

:$leftlbrace sum_\left\{i\right\}hat\left\{H\right\}_\left\{D\right\}\left(i\right) + sum_\left\{i>j\right\}frac\left\{1\right\}\left\{r_\left\{ij - sum_\left\{i>j\right\}hat\left\{B\right\}_\left\{ij\right\} ight brace Psi = E Psi$,

where

:: $hat\left\{H\right\}_\left\{D\right\}\left(i\right) = left \left[ q_\left\{i\right\}phi\left(mathbf\left\{r\right\}_\left\{i\right\}\right) + csum_\left\{s=x,y,z\right\}alpha_\left\{s\right\}\left(i\right)pi_\left\{s\right\}\left(I\right) + alpha_\left\{0\right\}\left(I\right)m_\left\{0\right\}c^\left\{2\right\} ight\right]$

is the Dirac hamiltonian (see Dirac equation) for particle "i" at position "r"i 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):

:$hat\left\{B\right\}_\left\{ij\right\} = -frac\left\{1\right\}\left\{2r_\left\{ij left \left[ mathbf\left\{a\right\}\left(i\right)cdotmathbf\left\{a\right\}\left(j\right) + frac\left\{ left\left(mathbf\left\{a\right\}\left(i\right)cdotmathbf\left\{r\right\}_\left\{ij\right\} ight\right) left\left(mathbf\left\{a\right\}\left(j\right)cdotmathbf\left\{r\right\}_\left\{ij\right\} ight\right) \right\}\left\{r_\left\{ij\right\}^\left\{2 ight\right]$,

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 " 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.

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) 1, which have well-defined meanings in the interaction of molecules with magnetic fields (for instance for nuclear magnetic resonance):

:$hat\left\{B\right\}_\left\{ij\right\} = hat\left\{H\right\}_\left\{0\right\} + hat\left\{H\right\}_\left\{1\right\} + ... + hat\left\{H\right\}_\left\{6\right\}$,

in which the consequitive partial operators are:

* $hat\left\{H\right\}_\left\{0\right\} = sum_\left\{i\right\}frac\left\{hat\left\{p\right\}_\left\{i\right\}^\left\{2\left\{2m_\left\{i + V$ is the nonrelativistic hamiltonian ("m_{i}" is the stationary mass of particle "i").

* $hat\left\{H\right\}_\left\{1\right\} = -frac\left\{1\right\}\left\{8c^\left\{2sum_\left\{i\right\}frac\left\{hat\left\{p\right\}_\left\{i\right\}^\left\{4\left\{m_\left\{i\right\}^\left\{3$ is connected to the dependence of mass on velocity: $E_\left\{kin\right\}^\left\{2\right\} - left\left(m_0c^2 ight\right)^2 = m^2v^2c^2$.

* $hat\left\{H\right\}_\left\{2\right\} = - sum_\left\{i>j\right\} frac\left\{q_iq_j\right\}\left\{2r_\left\{ij\right\}m_im_jc^2\right\} left \left[ mathbf\left\{hat\left\{p_icdotmathbf\left\{hat\left\{p_j + frac\left\{r_\left\{ij\right\}\left(r_\left\{ij\right\}mathbf\left\{hat\left\{p_i\right)cdotmathbf\left\{hat\left\{p_j\right\}\left\{r_\left\{ij\right\}^2\right\} ight\right]$ 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).

* $hat\left\{H\right\}_3 = frac\left\{mu_B\right\}\left\{c\right\} sum_i frac\left\{1\right\}\left\{m_i\right\} mathbf\left\{s\right\}_icdotleft \left[ mathbf\left\{F\right\}\left(mathbf\left\{r\right\}_\left\{ij\right\}\right) imesmathbf\left\{hat\left\{p_i + sum_\left\{j > i\right\} frac\left\{2q_i\right\}\left\{r_\left\{ij\right\}^3\right\}mathbf\left\{r\right\}_\left\{ij\right\} imesmathbf\left\{hat\left\{p_j ight\right]$ is the classical interaction between the orbital magnetic moments (from the orbital motion of charge) and spin magnetic moments (also called spin-orbit interaction). 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.

* $hat\left\{H\right\}_4 = frac\left\{ih\right\}\left\{8 pi c^2\right\} sum_\left\{i\right\} frac\left\{q_i\right\}\left\{m_i^2\right\} mathbf\left\{hat\left\{p_icdotmathbf\left\{F\right\}\left(mathbf\left\{r\right\}_i\right)$ is a nonclassical term characteristic for Dirac theory, sometimes called the Darwin term.

* $hat\left\{H\right\}_5 = 4mu_B^2 sum_\left\{i>j\right\} leftlbrace -frac\left\{8pi\right\}\left\{3\right\} \left(mathbf\left\{s\right\}_icdotmathbf\left\{s\right\}_j\right)delta\left(mathbf\left\{r\right\}_\left\{ij\right\}\right) + frac\left\{1\right\}\left\{r_\left\{ij\right\}^3\right\}left \left[ mathbf\left\{s\right\}_icdotmathbf\left\{s\right\}_j - frac\left\{\left(mathbf\left\{s\right\}_icdotmathbf\left\{r\right\}_\left\{ij\right\}\right)\left(mathbf\left\{s\right\}_jcdotmathbf\left\{r\right\}_\left\{ij\right\}\right)\right\}\left\{r_\left\{ij\right\}^2\right\} ight\right] ight brace$ is the magnetic moment spin-spin interaction. The first term is called the contact interaction, 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.

* $hat\left\{H\right\}_6 = 2mu_B sum_\left\{i\right\} left \left[ mathbf\left\{H\right\}\left(mathbf\left\{r\right\}_i\right)cdotmathbf\left\{s\right\}_i + frac\left\{q_i\right\}\left\{m_ic\right\}mathbf\left\{A\right\}\left(mathbf\left\{r\right\}_i\right)cdotmathbf\left\{hat\left\{p_i ight\right]$ is the interaction between spin and orbital magnetic moments with an external magnetic field H.

ee also

References

*fnb|1 H.A. Bethe, E.E. Salpeter, "Quantum Mechanics of One- and Two-Electron Atoms", Plenum Press, New York 1977, pg.181

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Breit (disambiguation) — Breit may refer to: * Breit, a municipality in Rhineland Palatinate, Germany * Breit equation * Relativistic Breit Wigner distribution * Breit Rabi Oscillation , Rabi cycle Persons with the surname Breit * Franz Breit (1817 1868), Obstetrician *… …   Wikipedia

• Dirac equation — Quantum field theory (Feynman diagram) …   Wikipedia

• Gregory Breit — (July 14, 1899 ndash; September 11, 1981) was a Russian born American physicist and professor at universities in New York, Wisconsin, Yale, and Buffalo. Together with Eugene Wigner he gave a description of particle resonant states, and with… …   Wikipedia

• Relativistic Breit–Wigner distribution — The relativistic Breit–Wigner distribution (after Gregory Breit and Eugene Wigner) is a continuous probability distribution with the following probability density function: Where k is the constant of proportionality, equal to with …   Wikipedia

• Gregory Breit — Gregory Breit. Gregory Breit (14 juillet 1899 11 septembre 1981) était un physicien américain d origine russe. Il mesura l altitude des couches de l ionosphère à l aide d impulsions radioéléctri …   Wikipédia en Français

• Darwin Lagrangian — The Darwin Lagrangian (named after Charles Galton Darwin, grandson of the biologist) describes the interaction to order between two charged particles in a vacuum and is given by where the free particle Lagrangian is …   Wikipedia

• 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. Such equations are called relativistic wave equations. The first such equation was… …   Wikipedia

• Paul Dirac — Paul Adrien Maurice Dirac Born Paul Adrien Maurice Dirac 8 August 1902(1902 08 08) Bristol, England …   Wikipedia

• Cauchy distribution — Not to be confused with Lorenz curve. Cauchy–Lorentz Probability density function The purple curve is the standard Cauchy distribution Cumulative distribution function …   Wikipedia

• Martin David Kruskal — Born September 28, 1925(1925 09 28) New York City …   Wikipedia