 Solution

For kinetic and field energies small compared with $m{c}^2$, the minimally coupled lower spinor is $\tilde{\chi }\simeq (\mathbf{\sigma }\cdot \mathbf{\pi })\tilde{\phi }/(2mc)$. Inserting it into the upper equation and using $[{\pi }_i,{\pi }_j]=i(e\hslash /c){\epsilon }_{ijk}{B}_k$ gives the Pauli equationThe spin moment is $e\hslash /(2mc)$ multiplying $\mathbf{\sigma }$, or gyromagnetic factor two relative to physical spin $\hslash \mathbf{\sigma }/2$.

For the alternative equation, elimination givesTo leading kinetic order and leading magnetic interaction this yieldsKeeping the source electric coupling during the same elimination writes the kinetic part more explicitly asThis product includes derivatives acting on the field and must not be treated as an ordinary scalar product of commuting variables. The parameter $\kappa$ is the intrinsic magnetic-moment coefficient: the moment operator is $\kappa \mathbf{\sigma }$. As printed, the alternative equation has no minimal charge coupling, so it can describe a neutral particle carrying a magnetic moment. If a Pauli term were added to the charged equation instead, its coefficient would represent an additional, anomalous contribution to the charge-generated Dirac moment.