= Pauli equation
{c}
{title2=$i\hbar\partial_t\Psi=H_P\Psi$}
For a nonrelativistic spin-one-half electron of charge $-e$, with $e>0$, the Pauli Hamiltonian is
$$
H_P=\frac1{2m}(-i\hbar\nabla+eA)^2+\frac{e\hbar}{2m}\sigma\cdot B-e\phi.
$$
The wavefunction has two spin components and $B=\nabla\times A$. If $\pi=-i\hbar\nabla+eA$, then $[\pi_i,\pi_j]=-ie\hbar\epsilon_{ijk}B_k$; the <Pauli matrix multiplication law> gives $(\sigma\cdot\pi)^2=\pi^2+e\hbar\sigma\cdot B$, explaining the spin-coupling sign. Thus $i\hbar\partial_t\Psi=H_P\Psi$ extends the <Schrödinger equation> to include the electron's magnetic moment.
Back to article page
