Pauli equation

Pauli equation
In quantum mechanics, 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. 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.
Equation
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.
Weak magnetic fields
For the case of where the magnetic field is constant and homogenous, one may expand using the symmetric gauge , where is the position operator and A is now an operator. We obtain
where is the particle angular momentum operator and we neglected terms in the magnetic field squared . Therefore we obtain
where is the spin of the particle. The factor 2 in front of the spin is known as the Dirac g-factor. The term in , is of the form which is the usual interaction between a magnetic moment and a magnetic field, like in the Zeeman effect.
For an electron of charge in an isotropic constant magnetic field, one can further reduce the equation using the total angular momentum and Wigner-Eckart theorem. Thus we find
where is the Bohr magneton and is the magnetic quantum number related to . The term is known as the Landé g-factor, and is given here by
where is the orbital quantum number related to and is the total orbital quantum number related to .
From Dirac equation
The Pauli equation can be inferred from the non-relativistic limit of the Dirac equation, which is the relativistic quantum equation of motion for spin-½ particles.
Derivation
Dirac equation can be written as:
where and are two-component spinor, forming a bispinor.
Using the following ansatz:
with two new spinors , the equation becomes
In the non-relativistic limit, and the kinetic and electrostatic energies are small with respect to the rest energy .
Thus
Inserted in the upper component of Dirac equation, we find Pauli equation (general form):
From a Foldy–Wouthuysen transformation
The rigorous derivation of the Pauli equation follows from Dirac equation in an external field and performing a Foldy–Wouthuysen transformation 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 instead.
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
where is the four-momentum operator, is the electromagnetic four-potential, is proportional to the anomalous magnetic dipole moment, is the electromagnetic tensor, and are the Lorentzian spin matrices and the commutator of the gamma matrices . 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
Eponymous equations of physics
Quantum mechanics