Pauli equation

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. It was formulated by Wolfgang Pauli in 1927. For a particle of mass and electric charge , in an electromagnetic field described by the magnetic vector potential and the electric scalar potential , the Pauli equation reads: Here are the Pauli operators collected into a vector for convenience, and is the momentum operator in position representation. The state of the system, (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. 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 term for a free particle in the absence of an electromagnetic field is just where is the kinetic momentum, while in the presence of an electromagnetic field it involves the minimal coupling , where now is the kinetic momentum and is the canonical momentum. The Pauli operators can be removed from the kinetic energy term using the Pauli vector identity: Note that unlike a vector, the differential operator has non-zero cross product with itself. This can be seen by considering the cross product applied to a scalar function : where is the magnetic field. For the full Pauli equation, one then obtains 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.