Equation
For a particle of mass m and charge q, in an electromagnetic field described by the vector potential A = (Ax, Ay, Az) and scalar electric potential ϕ, the Pauli equation reads:
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Pauli equation (General)
where σ = (σx, σy, σz) are the Pauli matrices collected into a vector for convenience, p = −iħ∇ is the momentum operator wherein ∇ denotes the gradient operator, and
is the two-component spinor wavefunction, a column vector written in Dirac notation.
The Hamiltonian operator
is a 2 × 2 matrix operator, because of the Pauli matrices. 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 p2/2m where p is the kinetic momentum, while in the presence of an EM field we have the minimal coupling p = P − qA, where P is the canonical momentum.
The Pauli matrices can be removed from the kinetic energy term, using the Pauli vector identity:
to obtain
where B = ∇ × A is the magnetic field.
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