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atomic physics Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. Atomic physics typically refers to the study of atomic structure and the interaction between atoms. It is primarily concerned wit ...
, the electron magnetic moment, or more specifically the electron magnetic dipole moment, is the
magnetic moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electroma ...
of an
electron 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 n ...
resulting from its intrinsic properties of
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally ...
and
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons res ...
. The value of the electron magnetic moment is The electron magnetic moment has been measured to an accuracy of relative to the
Bohr magneton In atomic physics, the Bohr magneton (symbol ) is a physical constant and the natural unit for expressing the magnetic moment of an electron caused by its orbital or spin angular momentum. The Bohr magneton, in SI units is defined as \mu_\m ...
.


Magnetic moment of an electron

The electron is a
charged particle In physics, a charged particle is a particle with an electric charge. It may be an ion, such as a molecule or atom with a surplus or deficit of electrons relative to protons. It can also be an electron or a proton, or another elementary pa ...
with charge −, where is the unit of elementary charge. Its
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
comes from two types of rotation:
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally ...
and
orbital motion In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a ...
. From
classical electrodynamics Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model; It is, therefore, a classical fi ...
, a rotating distribution of
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons res ...
produces a
magnetic dipole In electromagnetism, a magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the size of the source is reduced to zero while keeping the magnetic moment constant. It is a magnetic analogue of the electric ...
, so that it behaves like a tiny
bar magnet A magnet is a material or object that produces a magnetic field. This magnetic field is invisible but is responsible for the most notable property of a magnet: a force that pulls on other ferromagnetic materials, such as iron, steel, nic ...
. One consequence is that an external
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
exerts a
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
on the electron
magnetic moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electroma ...
that depends on the orientation of this dipole with respect to the field. If the electron is visualized as a classical rigid body in which the mass and charge have identical distribution and motion that is rotating about an axis with
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
, its magnetic dipole moment is given by: \boldsymbol = \frac\,\mathbf\,, where e is the
electron 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 n ...
rest mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, i ...
. The
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
L in this equation may be the spin angular momentum, the orbital angular momentum, or the total angular momentum. The ratio between the true
spin magnetic moment In physics, mainly quantum mechanics and particle physics, a spin magnetic moment is the magnetic moment caused by the spin of elementary particles. For example, the electron is an elementary spin-1/2 fermion. Quantum electrodynamics gives the ...
and that predicted by this model is a
dimensionless A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
factor , known as the electron -factor: \boldsymbol = g_\text\,\frac\,\mathbf\,. It is usual to express the magnetic moment in terms of the
reduced Planck 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 ...
and the
Bohr magneton In atomic physics, the Bohr magneton (symbol ) is a physical constant and the natural unit for expressing the magnetic moment of an electron caused by its orbital or spin angular momentum. The Bohr magneton, in SI units is defined as \mu_\m ...
B: \boldsymbol = -g_\text\,\mu_\text\,\frac\,. Since the magnetic moment is quantized in units of B, correspondingly the angular momentum is quantized in units of .


Formal definition

Classical notions such as the center of charge and mass are, however, hard to make precise for a quantum elementary particle. In practice the definition used by experimentalists comes from the form factors F_i(q^2) appearing in the matrix element \langle p_f , j^\mu , p_i \rangle = \bar u(p_f) \left\ u(p_i) of the electromagnetic current operator between two on-shell states. Here u(p_i) and \bar u(p_f) are 4-spinor solution of the
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
normalized so that \bar u u=2m_, and q^\mu=p^\mu_f-p^\mu_i is the momentum transfer from the current to the electron. The q^2 = 0 form factor F_1(0) = -e is the electron's charge, \mu = ,F_1(0)+F_2(0)\, ,2\,m_\,/math> is its static magnetic dipole moment, and -F_3(0)/ ,2\,m_\,/math> provides the formal definion of the electron's electric dipole moment. The remaining form factor F_4(q^2) would, if non zero, be the anapole moment.


Spin magnetic dipole moment

The
spin magnetic moment In physics, mainly quantum mechanics and particle physics, a spin magnetic moment is the magnetic moment caused by the spin of elementary particles. For example, the electron is an elementary spin-1/2 fermion. Quantum electrodynamics gives the ...
is intrinsic for an electron. It is \boldsymbol_\text = -g_\,\mu_\text\,\frac\,. Here is the electron spin angular momentum. The spin -factor is approximately two: g_ \approx 2. The magnetic moment of an electron is approximately twice what it should be in classical mechanics. The factor of two implies that the electron appears to be twice as effective in producing a magnetic moment as the corresponding classical charged body. The spin magnetic dipole moment is approximately one B because g_ \approx 2 and the electron is a spin- particle (): \mu_ \approx 2\,\frac\frac = \mu_\text\,. The component of the electron magnetic moment is (\boldsymbol_\text)_z = -g_\text\,\mu_\text\,m_\text\,, where s is the
spin quantum number In atomic physics, the spin quantum number is a quantum number (designated ) which describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particle. The phrase was originally used to describe t ...
. Note that is a ''negative'' constant multiplied by the
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally ...
, so the magnetic moment is '' antiparallel'' to the spin angular momentum. The spin ''g''-factor comes from the
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
, a fundamental equation connecting the electron's spin with its electromagnetic properties. Reduction of the Dirac equation for an electron in a magnetic field to its non-relativistic limit yields the Schrödinger equation with a correction term, which takes account of the interaction of the electron's intrinsic magnetic moment with the magnetic field giving the correct energy. For the electron spin, the most accurate value for the spin -factor has been experimentally determined to have the value : Note that this differs only marginally from the value from the Dirac equation. The small correction is known as the
anomalous magnetic dipole moment In quantum electrodynamics, the anomalous magnetic moment of a particle is a contribution of effects of quantum mechanics, expressed by Feynman diagrams with loops, to the magnetic moment of that particle. (The ''magnetic moment'', also called '' ...
of the electron; it arises from the electron's interaction with virtual photons in
quantum electrodynamics In particle physics, quantum electrodynamics (QED) 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 spec ...
. A triumph of the
quantum electrodynamics In particle physics, quantum electrodynamics (QED) 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 spec ...
theory is the accurate prediction of the electron ''g''-factor. The CODATA value for the electron magnetic moment is :


Orbital magnetic dipole moment

The revolution of an electron around an axis through another object, such as the nucleus, gives rise to the orbital magnetic dipole moment. Suppose that the angular momentum for the orbital motion is . Then the orbital magnetic dipole moment is \boldsymbol_L = -g_\text\,\mu_\text\,\frac\,. Here L is the electron orbital -factor and B is the
Bohr magneton In atomic physics, the Bohr magneton (symbol ) is a physical constant and the natural unit for expressing the magnetic moment of an electron caused by its orbital or spin angular momentum. The Bohr magneton, in SI units is defined as \mu_\m ...
. The value of L is exactly equal to one, by a quantum-mechanical argument analogous to the derivation of the classical gyromagnetic ratio.


Total magnetic dipole moment

The total
magnetic dipole moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagnet ...
resulting from both spin and orbital angular momenta of an electron is related to the total angular momentum by a similar equation: \boldsymbol_\text = -g_\text\,\mu_\text\,\frac\,. The -factor J is known as the Landé ''g''-factor, which can be related to L and S by quantum mechanics. See Landé ''g''-factor for details.


Example: hydrogen atom

For a
hydrogen Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-to ...
atom, an
electron 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 n ...
occupying the
atomic orbital In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any ...
 , the
magnetic dipole moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagnet ...
is given by :\mu_\text = -g_\text \frac\langle\Psi_, L, \Psi_\rangle = -\mu_\text\sqrt. Here is the orbital
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
, , , and are the principal,
azimuthal An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north. Mathematically, ...
, and
magnetic Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particl ...
quantum numbers In quantum physics and chemistry, quantum numbers describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be k ...
respectively. The component of the orbital magnetic dipole moment for an electron with a
magnetic quantum number In atomic physics, the magnetic quantum number () is one of the four quantum numbers (the other three being the principal, azimuthal, and spin) which describe the unique quantum state of an electron. The magnetic quantum number distinguishes the ...
is given by :(\boldsymbol_\text)_z = -\mu_\text m_\ell.


History

The electron magnetic moment is intrinsically connected to electron spin and was first hypothesized during the early models of the atom in the early twentieth century. The first to introduce the idea of electron spin was
Arthur Compton Arthur Holly Compton (September 10, 1892 – March 15, 1962) was an American physicist who won the Nobel Prize in Physics in 1927 for his 1923 discovery of the Compton effect, which demonstrated the particle nature of electromagnetic radia ...
in his 1921 paper on investigations of ferromagnetic substances with X-rays. In Compton's article, he wrote: "Perhaps the most natural, and certainly the most generally accepted view of the nature of the elementary magnet, is that the revolution of electrons in orbits within the atom give to the atom as a whole the properties of a tiny permanent magnet." That same year
Otto Stern :''Otto Stern was also the pen name of German women's rights activist Louise Otto-Peters (1819–1895)''. Otto Stern (; 17 February 1888 – 17 August 1969) was a German-American physicist and Nobel laureate in physics. He was the second most ...
proposed an experiment carried out later called the
Stern–Gerlach experiment The Stern–Gerlach experiment demonstrated that the spatial orientation of angular momentum is quantized. Thus an atomic-scale system was shown to have intrinsically quantum properties. In the original experiment, silver atoms were sent throug ...
in which silver atoms in a magnetic field were deflected in opposite directions of distribution. This pre-1925 period marked the
old quantum theory The old quantum theory is a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was rather a set of heuristic corrections to classical mechanics. The theory ...
built upon the Bohr-Sommerfeld model of the atom with its classical elliptical electron orbits. During the period between 1916 and 1925, much progress was being made concerning the arrangement of electrons in the
periodic table The periodic table, also known as the periodic table of the (chemical) elements, is a rows and columns arrangement of the chemical elements. It is widely used in chemistry, physics, and other sciences, and is generally seen as an icon of ch ...
. In order to explain the
Zeeman effect The Zeeman effect (; ) is the effect of splitting of a spectral line into several components in the presence of a static magnetic field. It is named after the Dutch physicist Pieter Zeeman, who discovered it in 1896 and received a Nobel pr ...
in the Bohr atom, Sommerfeld proposed that electrons would be based on three 'quantum numbers', n, k, and m, that described the size of the orbit, the shape of the orbit, and the direction in which the orbit was pointing.
Irving Langmuir Irving Langmuir (; January 31, 1881 – August 16, 1957) was an American chemist, physicist, and engineer. He was awarded the Nobel Prize in Chemistry in 1932 for his work in surface chemistry. Langmuir's most famous publication is the 1919 ar ...
had explained in his 1919 paper regarding electrons in their shells, "Rydberg has pointed out that these numbers are obtained from the series N = 2(1 + 2^2 + 2^2 + 3^2 + 3^2 + 4^2). The factor two suggests a fundamental two-fold symmetry for all stable atoms." This 2n^2 configuration was adopted by Edmund Stoner, in October 1924 in his paper 'The Distribution of Electrons Among Atomic Levels' published in the Philosophical Magazine.
Wolfgang Pauli Wolfgang Ernst Pauli (; ; 25 April 1900 – 15 December 1958) was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after having been nominated by Albert Einstein, Pauli received the Nobel Prize in Physics ...
hypothesized that this required a fourth quantum number with a two-valuedness.


Electron spin in the Pauli and Dirac theories

Starting from here the charge of the electron is  . The necessity of introducing half-integral
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally ...
goes back experimentally to the results of the
Stern–Gerlach experiment The Stern–Gerlach experiment demonstrated that the spatial orientation of angular momentum is quantized. Thus an atomic-scale system was shown to have intrinsically quantum properties. In the original experiment, silver atoms were sent throug ...
. A beam of atoms is run through a strong non-uniform magnetic field, which then splits into parts depending on the intrinsic angular momentum of the atoms. It was found that for
silver Silver is a chemical element with the symbol Ag (from the Latin ', derived from the Proto-Indo-European ''h₂erǵ'': "shiny" or "white") and atomic number 47. A soft, white, lustrous transition metal, it exhibits the highest electrical ...
atoms, the beam was split in two—the ground state therefore could not be integral, because even if the intrinsic angular momentum of the atoms were as small as possible, 1, the beam would be split into 3 parts, corresponding to atoms with = −1, 0, and +1. The conclusion is that silver atoms have net intrinsic angular momentum of .
Pauli Pauli is a surname and also a Finnish male given name (variant of Paul) and may refer to: * Arthur Pauli (born 1989), Austrian ski jumper * Barbara Pauli (1752 or 1753 - fl. 1781), Swedish fashion trader *Gabriele Pauli (born 1957), German politi ...
set up a theory which explained this splitting by introducing a two-component wave function and a corresponding correction term in the Hamiltonian, representing a semi-classical coupling of this
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
to an applied magnetic field, as so: :H = \frac \left \boldsymbol\cdot \left ( \mathbf - \frac\mathbf \right ) \right 2 + e\phi. Here is the
magnetic vector potential In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its curl is equal to the magnetic field: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the magnetic ...
and the
electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
, both representing the
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...
, and = (, , ) are the
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used ...
. On squaring out the first term, a residual interaction with the magnetic field is found, along with the usual classical Hamiltonian of a charged particle interacting with an applied field: :H = \frac\left ( \mathbf - \frac\mathbf \right )^2 + e\phi - \frac\boldsymbol\cdot \mathbf. This Hamiltonian is now a 2 × 2 matrix, so the Schrödinger equation based on it must use a two-component wave function. Pauli had introduced the 2 × 2 sigma matrices as pure ''phenomenology'' — Dirac now had a ''theoretical argument'' that implied that
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally ...
was somehow the consequence of incorporating relativity into
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, ...
. On introducing the external electromagnetic 4-potential into the Dirac equation in a similar way, known as minimal coupling, it takes the form (in
natural units In physics, natural units are physical units of measurement in which only universal physical constants are used as defining constants, such that each of these constants acts as a coherent unit of a quantity. For example, the elementary charge ma ...
= = 1) :\left -i\gamma^\mu\left ( \partial_\mu + ieA_\mu \right ) + m \right \psi = 0\, where \scriptstyle \gamma^\mu are the
gamma matrices In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\ma ...
(known as
Dirac matrices In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\mat ...
) and is the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
. A second application of the
Dirac operator In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise form ...
will now reproduce the Pauli term exactly as before, because the spatial Dirac matrices multiplied by , have the same squaring and commutation properties as the Pauli matrices. What is more, the value of the
gyromagnetic ratio In physics, the gyromagnetic ratio (also sometimes known as the magnetogyric ratio in other disciplines) of a particle or system is the ratio of its magnetic moment to its angular momentum, and it is often denoted by the symbol , gamma. Its SI u ...
of the electron, standing in front of Pauli's new term, is explained from first principles. This was a major achievement of the Dirac equation and gave physicists great faith in its overall correctness. The Pauli theory may be seen as the low energy limit of the Dirac theory in the following manner. First the equation is written in the form of coupled equations for 2-spinors with the units restored: :\begin (mc^2 - E + e \phi) & c\sigma\cdot \left (\mathbf - \frac\mathbf \right ) \\ -c\boldsymbol\cdot \left ( \mathbf - \frac\mathbf \right ) & \left ( mc^2 + E - e \phi \right ) \end \begin \psi_+ \\ \psi_- \end = \begin 0 \\ 0 \end. so :\begin (E - e\phi) \psi_+ - c\boldsymbol \cdot \left( \mathbf - \frac\mathbf \right) \psi_- &= mc^2 \psi_+ \\ -(E - e\phi) \psi_- + c\boldsymbol \cdot \left( \mathbf - \frac\mathbf \right) \psi_+ &= mc^2 \psi_- \end Assuming the field is weak and the motion of the electron non-relativistic, we have the total energy of the electron approximately equal to its
rest energy The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
, and the momentum reducing to the classical value, :\begin E - e\phi &\approx mc^2 \\ p &\approx m v \end and so the second equation may be written :\psi_- \approx \frac \boldsymbol \cdot \left( \mathbf - \frac\mathbf \right) \psi_+ which is of order - thus at typical energies and velocities, the bottom components of the
Dirac spinor In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain co ...
in the standard representation are much suppressed in comparison to the top components. Substituting this expression into the first equation gives after some rearrangement : \left(E - mc^2\right) \psi_+ = \frac \left \boldsymbol\cdot \left( \mathbf - \frac\mathbf \right) \right2 \psi_+ + e\phi \psi_+ The operator on the left represents the particle energy reduced by its rest energy, which is just the classical energy, so we recover Pauli's theory if we identify his 2-spinor with the top components of the Dirac spinor in the non-relativistic approximation. A further approximation gives the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
as the limit of the Pauli theory. Thus the Schrödinger equation may be seen as the far non-relativistic approximation of the Dirac equation when one may neglect spin and work only at low energies and velocities. This also was a great triumph for the new equation, as it traced the mysterious that appears in it, and the necessity of a complex wave function, back to the geometry of space-time through the Dirac algebra. It also highlights why the Schrödinger equation, although superficially in the form of a diffusion equation, actually represents the propagation of waves. It should be strongly emphasized that this separation of the Dirac spinor into large and small components depends explicitly on a low-energy approximation. The entire Dirac spinor represents an ''irreducible'' whole, and the components we have just neglected to arrive at the Pauli theory will bring in new phenomena in the relativistic regime -
antimatter In modern physics, antimatter is defined as matter composed of the antiparticles (or "partners") of the corresponding particles in "ordinary" matter. Antimatter occurs in natural processes like cosmic ray collisions and some types of radioac ...
and the idea of creation and annihilation of particles. In a general case (if a certain linear function of electromagnetic field does not vanish identically), three out of four components of the spinor function in the Dirac equation can be algebraically eliminated, yielding an equivalent fourth-order partial differential equation for just one component. Furthermore, this remaining component can be made real by a gauge transform.


Measurement

The existence of the anomalous magnetic moment of the electron has been detected experimentally by
magnetic resonance Magnetic resonance is a process by which a physical excitation (resonance) is set up via magnetism. This process was used to develop magnetic resonance imaging and Nuclear magnetic resonance spectroscopy technology. It is also being used to d ...
method. This allows the determination of
hyperfine splitting In atomic physics, hyperfine structure is defined by small shifts in otherwise degenerate energy levels and the resulting splittings in those energy levels of atoms, molecules, and ions, due to electromagnetic multipole interaction between the ...
of electron shell energy levels in atoms of protium and
deuterium Deuterium (or hydrogen-2, symbol or deuterium, also known as heavy hydrogen) is one of two stable isotopes of hydrogen (the other being protium, or hydrogen-1). The nucleus of a deuterium atom, called a deuteron, contains one proton and one ...
using the measured resonance frequency for several transitions. The
magnetic moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electroma ...
of the electron has been measured using a one-electron quantum
cyclotron A cyclotron is a type of particle accelerator invented by Ernest O. Lawrence in 1929–1930 at the University of California, Berkeley, and patented in 1932. Lawrence, Ernest O. ''Method and apparatus for the acceleration of ions'', filed: Jan ...
and quantum nondemolition spectroscopy. The spin frequency of the electron is determined by the -factor. : \nu_s = \frac \nu_c : \frac = \frac


See also

*
Spin (physics) Spin is a conserved quantity carried by elementary particles, and thus by composite particles (hadrons) and atomic nuclei. Spin is one of two types of angular momentum in quantum mechanics, the other being ''orbital angular momentum''. The orbit ...
*
Electron precipitation Electron precipitation (also called energetic electron precipitation or EEP) is an atmospheric phenomenon that occurs when previously trapped electrons enter the Earth's atmosphere, thus creating communications interferences and other disturbances. ...
*
Bohr magneton In atomic physics, the Bohr magneton (symbol ) is a physical constant and the natural unit for expressing the magnetic moment of an electron caused by its orbital or spin angular momentum. The Bohr magneton, in SI units is defined as \mu_\m ...
* Nuclear magnetic moment *
Nucleon magnetic moment The nucleon magnetic moments are the intrinsic magnetic dipole moments of the proton and neutron, symbols ''μ''p and ''μ''n. Protons and neutrons, both nucleons, comprise the nucleus of atoms, and both nucleons behave as small magnets whose st ...
*
Anomalous magnetic dipole moment In quantum electrodynamics, the anomalous magnetic moment of a particle is a contribution of effects of quantum mechanics, expressed by Feynman diagrams with loops, to the magnetic moment of that particle. (The ''magnetic moment'', also called '' ...
*
Electron electric dipole moment The electron electric dipole moment is an intrinsic property of an electron such that the potential energy is linearly related to the strength of the electric field: :U = \mathbf d_ \cdot \mathbf E. The electron's electric dipole moment (EDM) m ...
*
Fine structure In atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the non-relativistic Schrödinger equation. It was first measured precisely for the hydrogen atom ...
*
Hyperfine structure In atomic physics, hyperfine structure is defined by small shifts in otherwise degenerate energy levels and the resulting splittings in those energy levels of atoms, molecules, and ions, due to electromagnetic multipole interaction between the n ...


References


Bibliography

* * {{DEFAULTSORT:Electron Magnetic Dipole Moment Atomic physics Electric dipole moment Magnetic moment Particle physics Physical constants