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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, q ...
, the Pauli equation or Schrödinger–Pauli equation is the formulation of the Schrödinger equation for spin-½ particles, which takes into account the interaction of the particle's spin 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 classica ...
. It is the non- relativistic limit of the Dirac equation 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 fo ...
, so that relativistic effects can be neglected. It was formulated by Wolfgang Pauli 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 classica ...
described by the magnetic vector potential \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 i ...
\phi, the Pauli equation reads: Here \boldsymbol = (\sigma_x, \sigma_y, \sigma_z) are the
Pauli operators 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 ...
collected into a vector for convenience, and \mathbf = -i\hbar \nabla is the momentum operator in position representation. The state of the system, , \psi\rangle (written in Dirac notation), can be considered as a two-component spinor wavefunction, 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, ...
(after choice of basis): : , \psi\rangle = \psi_+ , \mathord\uparrow\rangle + \psi_-, \mathord\downarrow\rangle \,\stackrel\, \begin \psi_+ \\ \psi_- \end. The Hamiltonian operator is a 2 × 2 matrix because of the
Pauli operators 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 ...
. :\hat = \frac \left boldsymbol\cdot(\mathbf - q \mathbf) \right2 + q \phi Substitution into the Schrödinger equation gives the Pauli equation. This Hamiltonian is similar to the classical Hamiltonian for a charged particle interacting with an electromagnetic field. See Lorentz force 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 a ...
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 \mathbf = \mathbf - q\mathbf, where now \mathbf is the kinetic momentum and \mathbf is the canonical momentum. 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 sy ...
operator and we neglected terms in the magnetic field squared B^2. Therefore we obtain
where \mathbf=\hbar\boldsymbol/2 is the spin 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. 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 and m_j is the magnetic quantum number related to \mathbf. The term g_J is known as the Landé g-factor, and is given here by :g_J = \frac+\frac, where \ell is the orbital quantum number 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, 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, 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.


Pauli coupling

Pauli's equation is derived by requiring minimal coupling, 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, 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 operator, A_\mu is the electromagnetic four-potential, a is proportional to the anomalous magnetic dipole moment, F^=\partial^A^-\partial^A^ is the electromagnetic tensor, and \sigma_=\frac gamma_,\gamma_/math> are the Lorentzian spin matrices and the commutator of the gamma matrices \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) for an arbitrary ''g''-factor.


See also

* Semiclassical physics * Atomic, molecular, and optical physics * Group contraction * Gordon decomposition


Footnotes


References


Books

* * * {{DEFAULTSORT:Pauli Equation Quantum mechanics