Pauli equation

For a nonrelativistic spin-one-half electron of charge $-e$, with $e>0$, the Pauli Hamiltonian isThe wavefunction has two spin components and $B=\nabla \times A$. If $\pi =-i\hslash \nabla +eA$, then $[{\pi }_i,{\pi }_j]=-ie\hslash {ϵ}_{ijk}{B}_k$; the Pauli matrix multiplication law gives $(\sigma \cdot \pi {)}^2={\pi }^2+e\hslash \sigma \cdot B$, explaining the spin-coupling sign. Thus $i\hslash {\partial }_t\Psi =H_P\Psi$ extends the Schrödinger equation to include the electron's magnetic moment.