Pauli equation

In quantum mechanics, the Pauli equation, or Schrödinger–Pauli equation, is the formulation of the Schrodinger 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.

[\frac{1}{2 m} (σ \cdot (\hat{p} - q A))^{2} + q ϕ] | ψ ⟩ = i ℏ \frac{\partial}{\partial t} | ψ ⟩

Here $σ = (σ_{x}, σ_{y}, σ_{z})$ are the Pauli operators collected into a vector for convenience, and $\hat{p} = - i ℏ \nabla$ is the momentum operator in position representation. The state of the system, $| ψ ⟩$ , can be considered as a two-component spinor wavefunction, or a column vector (after choice of basis):

| ψ ⟩ = ψ_{+} | ↑ ⟩ + ψ_{-} | ↓ ⟩ = (\begin{matrix} ψ_{+} \\ ψ_{-} \end{matrix}) .

The Hamiltonian operator is a $2 \times 2$ matrix because of the Pauli operators:

\hat{H} = \frac{1}{2 m} {[σ \cdot (\hat{p} - q A)]}^{2} + q ϕ .