In quantum mechanics, the Pauli equation or Schrödinger–Pauli equation is the formulation of the Schrödinger equation for spin-1/2 particles, which takes into account the interaction of the particle's spin with an external electromagnetic field.
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, so that relativistic effects can be neglected.
, in an electromagnetic field described by the magnetic vector potential
are the Pauli operators collected into a vector for convenience, and
(written in Dirac notation), can be considered as a two-component spinor wavefunction, or a column vector (after choice of basis): The Hamiltonian operator is a 2 × 2 matrix because of the Pauli operators.
See Lorentz force for details of this classical case.
The kinetic energy term for a free particle in the absence of an electromagnetic field is just
is the kinetic momentum, while in the presence of an electromagnetic field it involves the minimal coupling
The Pauli operators can be removed from the kinetic energy term using the Pauli vector identity: Note that unlike a vector, the differential operator
This can be seen by considering the cross product applied to a scalar function
For the full Pauli equation, one then obtains[2]
for which only a few analytic results are known, e.g., in the context of Landau quantization with homogenous magnetic fields or for an idealized, Coulomb-like, inhomogeneous magnetic field.
[3] For the case of where the magnetic field is constant and homogenous, one may expand
is the particle angular momentum operator and we neglected terms in the magnetic field squared
The factor 2 in front of the spin is known as the Dirac g-factor.
which is the usual interaction between a magnetic moment
and a magnetic field, like in the Zeeman effect.
in an isotropic constant magnetic field, one can further reduce the equation using the total angular momentum
is the magnetic quantum number related to
is the orbital quantum number related to
is the total orbital quantum number related to
are two-component spinor, forming a bispinor.
Inserted in the upper component of Dirac equation, we find Pauli equation (general form):
The rigorous derivation of the Pauli equation follows from Dirac equation in an external field and performing a Foldy–Wouthuysen transformation[4] considering terms up to order
Similarly, higher order corrections to the Pauli equation can be determined giving rise to spin-orbit and Darwin interaction terms, when expanding up to order
[6] 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 where
is proportional to the anomalous magnetic dipole moment,
[7][8] 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.