Intuitive

The Pauli equation describes how the state of a quantum system with half-integer spin changes in time.

In contrast, the Schrödinger equation describes the time evolution of systems without spin.

Concrete

The Pauli equation is the non-relativistic limit of the Dirac equation.

Abstract

• Nonrelativistic particles and wave equations by Jean-Marc Lévy-Leblond

Why is it interesting?

The Pauli equation is the correct non-relativistic equation to describe spin $1/2$ particles.

Definitions

• $\Psi$ is the wave function,
• $m$ the mass of the particle,
• $q$ the charge of the particle,
• $\vec{\sigma}$ the Pauli matrices,
• $\vec{p}$ the momentum,
• $\vec A$ the vector potential,
• $\phi$ the electric scalar potential and
• $\hbar$ the reduced Planck constant.

Take note that $\vec \sigma$, a "vector of matrices" is only used as a convenient short-hand notation for the sums that appear in the equation. For example, $\vec \sigma \vec p = \sigma_1 p_1 + \sigma_2 p_2 + \sigma_3 p_3.$