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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, ...
, the Pauli equation or Schrödinger–Pauli equation is the formulation of 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 the ...
for
spin-½ In quantum mechanics, spin is an intrinsic property of all elementary particles. All known fermions, the particles that constitute ordinary matter, have a spin of . The spin number describes how many symmetrical facets a particle has in one ful ...
particles, which takes into account the interaction of the particle's
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 b ...
with an external
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 c ...
. It is the non- relativistic limit 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 ...
and can be used where particles are moving at speeds much less than the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
, so that relativistic effects can be neglected. It was formulated by
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 fo ...
in 1927.


Equation

For a particle of mass m and electric charge q, in an
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 c ...
described by 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 v ...
\mathbf and the
electric scalar 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 ...
\phi, the Pauli equation reads: Here \boldsymbol = (\sigma_x, \sigma_y, \sigma_z) are the Pauli operators collected into a vector for convenience, and \mathbf = -i\hbar \nabla is the
momentum operator In quantum mechanics, the momentum operator is the operator (physics), operator associated with the momentum (physics), linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case o ...
in position representation. The state of the system, , \psi\rangle (written in
Dirac notation Distributed Research using Advanced Computing (DiRAC) is an integrated supercomputing facility used for research in particle physics, astronomy and cosmology in the United Kingdom. DiRAC makes use of multi-core processors and provides a variety of ...
), can be considered as a two-component
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
wavefunction 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 mad ...
, or a
column vector In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, c ...
(after choice of basis): : , \psi\rangle = \psi_+ , \mathord\uparrow\rangle + \psi_-, \mathord\downarrow\rangle \,\stackrel\, \begin \psi_+ \\ \psi_- \end. The
Hamiltonian operator 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 Hamiltonia ...
is a 2 × 2 matrix because of the Pauli operators. :\hat = \frac \left boldsymbol\cdot(\mathbf - q \mathbf) \right2 + q \phi Substitution into 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 the ...
gives the Pauli equation. This Hamiltonian is similar to the classical Hamiltonian for a charged particle interacting with an electromagnetic field. See
Lorentz force In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...
for details of this classical case. The
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
term for a free particle in the absence of an electromagnetic field is just \frac where \mathbf is the ''kinetic'' momentum, while in the presence of an electromagnetic field it involves the
minimal coupling In analytical mechanics and quantum field theory, minimal coupling refers to a coupling between field theory (physics), fields which involves only the electric charge, charge distribution and not higher multipole moments of the charge distribution. ...
\mathbf = \mathbf - q\mathbf, where now \mathbf is the
kinetic momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass and ...
and \mathbf is the
canonical momentum In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of cla ...
. The Pauli operators can be removed from the kinetic energy term using the Pauli vector identity: :(\boldsymbol\cdot \mathbf)(\boldsymbol\cdot \mathbf) = \mathbf\cdot\mathbf + i\boldsymbol\cdot \left(\mathbf \times \mathbf\right) Note that unlike a vector, the differential operator \mathbf - q\mathbf = -i \hbar \nabla - q \mathbf has non-zero cross product with itself. This can be seen by considering the cross product applied to a scalar function \psi: :\left left(\mathbf - q\mathbf\right) \times \left(\mathbf - q\mathbf\right)\rightpsi = -q \left mathbf \times \left(\mathbf\psi\right) + \mathbf \times \left(\mathbf\psi\right)\right= i q \hbar \left nabla \times \left(\mathbf\psi\right) + \mathbf \times \left(\nabla\psi\right)\right= i q \hbar \left psi\left(\nabla \times \mathbf\right) - \mathbf \times \left(\nabla\psi\right) + \mathbf \times \left(\nabla\psi\right)\right= i q \hbar \mathbf \psi where \mathbf = \nabla \times \mathbf is the magnetic field. For the full Pauli equation, one then obtains


Weak magnetic fields

For the case of where the magnetic field is constant and homogenous, one may expand (\mathbf-q\mathbf)^2 using the symmetric gauge \mathbf=\frac\mathbf\times\mathbf, where \mathbf is the position operator and A is now an operator. We obtain :(\mathbf \hat-q \mathbf \hat)^2 = , \mathbf, ^ - q(\mathbf\times\mathbf \hat)\cdot \mathbf +\fracq^2\left(, \mathbf, ^2, \mathbf, ^2-, \mathbf\cdot\mathbf, ^2\right) \approx \mathbf^ - q\mathbf \hat\cdot\mathbf B\,, where \mathbf is the particle
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 ...
operator and we neglected terms in the magnetic field squared B^2. Therefore we obtain
where \mathbf=\hbar\boldsymbol/2 is 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 b ...
of the particle. The factor 2 in front of the spin is known as the Dirac ''g''-factor. The term in \mathbf, is of the form -\boldsymbol\cdot\mathbf which is the usual interaction between a magnetic moment \boldsymbol and a magnetic field, like in 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 prize ...
. For an electron of charge -e in an isotropic constant magnetic field, one can further reduce the equation using the total angular momentum \mathbf=\mathbf+\mathbf and Wigner-Eckart theorem. Thus we find : \left \mathbf, - e \phi\right\psi\rangle = i \hbar \frac , \psi\rangle where \mu_=\frac 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_\mathrm ...
and m_j is the
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 th ...
related to \mathbf. The term g_J is known as the
Landé g-factor In physics, the Landé ''g''-factor is a particular example of a ''g''-factor, namely for an electron with both spin and orbital angular momenta. It is named after Alfred Landé, who first described it in 1921. In atomic physics, the Landé '' ...
, and is given here by :g_J = \frac+\frac, where \ell is the
orbital quantum number The azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers that describe t ...
related to L^2 and j is the total orbital quantum number related to J^2.


From Dirac equation

The Pauli equation is the non-relativistic limit of
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 ...
, the relativistic quantum equation of motion for particles spin-½.


Derivation

Dirac equation can be written as: i \hbar\, \partial_t \begin \psi_1 \\ \psi_2\end = c \, \begin \boldsymbol\cdot \boldsymbol \Pi \,\psi_2 \\ \boldsymbol\cdot \boldsymbol \Pi \,\psi_1\end + q\, \phi \, \begin \psi_1 \\ \psi_2\end + mc^2\, \begin \psi_1 \\ -\psi_2\end , where \partial_t=\frac and \psi_1,\psi_2 are two-component
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
, forming a bispinor. Using the following ansatz: \begin \psi_1 \\ \psi_2 \end = e^ \begin \psi \\ \chi \end , with two new spinors \psi,\chi, the equation becomes i \hbar \partial_t \begin \psi \\ \chi\end = c \, \begin \boldsymbol\cdot \boldsymbol \Pi \,\chi\\ \boldsymbol\cdot \boldsymbol \Pi \,\psi\end +q\, \phi \, \begin \psi\\ \chi \end + \begin 0 \\ -2\,mc^2\, \chi \end . In the non-relativistic limit, \partial_t \chi and the kinetic and electrostatic energies are small with respect to the rest energy mc^2. Thus \chi \approx \frac\,. Inserted in the upper component of Dirac equation, we find Pauli equation (general form): i \hbar\, \partial_t \, \psi= \left frac +q\, \phi\right\psi.


From a Foldy–Wouthuysen transformation

One can also rigorously derive Pauli equation, starting from Dirac equation in an external field and performing a
Foldy–Wouthuysen transformation The Foldy–Wouthuysen transformation was historically significant and was formulated by Leslie Lawrance Foldy and Siegfried Adolf Wouthuysen in 1949 to understand the nonrelativistic limit of the Dirac equation, the equation for spin-½ partic ...
.


Pauli coupling

Pauli's equation is derived by requiring
minimal coupling In analytical mechanics and quantum field theory, minimal coupling refers to a coupling between field theory (physics), fields which involves only the electric charge, charge distribution and not higher multipole moments of the charge distribution. ...
, which provides a ''g''-factor ''g''=2. Most elementary particles have anomalous ''g''-factors, different from 2. In the domain of relativistic
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, one defines a non-minimal coupling, sometimes called Pauli coupling, in order to add an anomalous factor :\gamma^p_\mu\to \gamma^p_\mu-q\gamma^A_\mu +a\sigma_F^ where p_\mu is the
four-momentum In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is ...
operator, A_\mu is the
electromagnetic four-potential An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.Gravitation, J.A. Whe ...
, a is proportional to 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 '' ...
, F^=\partial^A^-\partial^A^ is the
electromagnetic tensor In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. T ...
, and \sigma_=\frac gamma_,\gamma_/math> are the Lorentzian spin matrices and the commutator of 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 ...
\gamma^. In the context of non-relativistic quantum mechanics, instead of working with the Schrödinger equation, Pauli coupling is equivalent to using the Pauli equation (or postulating
Zeeman energy Zeeman energy, or the external field energy, is the potential energy of a magnetised body in an external magnetic field. It is named after the Dutch physicist Pieter Zeeman, primarily known for the Zeeman effect. In SI units, it is given by :E_ = ...
) for an arbitrary ''g''-factor.


See also

*
Semiclassical physics Semiclassical physics, or simply semiclassical refers to a theory in which one part of a system is described quantum mechanically whereas the other is treated classically. For example, external fields will be constant, or when changing will be ...
*
Atomic, molecular, and optical physics Atomic, molecular, and optical physics (AMO) is the study of matter-matter and light-matter interactions; at the scale of one or a few atoms and energy scales around several electron volts. The three areas are closely interrelated. AMO theory in ...
*
Group contraction In theoretical physics, Eugene Wigner and Erdal İnönü have discussed the possibility to obtain from a given Lie group a different (non-isomorphic) Lie group by a group contraction with respect to a continuous subgroup of it. That amounts to a ...
*
Gordon decomposition In mathematical physics, the Gordon decomposition (named after Walter Gordon) of the Dirac current is a splitting of the charge or particle-number current into a part that arises from the motion of the center of mass of the particles and a part tha ...


Footnotes


References


Books

* * * {{DEFAULTSORT:Pauli Equation Quantum mechanics