$\left( \frac{1}{2m}(\vec \sigma ( \vec p - q\vec A))^2 + q\phi \right) \Psi = i \hbar \partial_t \Psi $ ====== Pauli Equation ====== The Pauli equation describes how the state of a quantum system with [[basic_notions:spin|half-integer spin]] changes in time. In contrast, the [[equations:schroedinger_equation|Schrödinger equation]] describes the time evolution of systems without spin. The Pauli equation is the non-relativistic limit of the [[equations:dirac_equation|Dirac equation]]. * Nonrelativistic particles and wave equations by Jean-Marc Lévy-Leblond The Pauli equation is the correct __non-relativistic__ equation to describe spin $1/2$ particles. * $\Psi$ 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. $