The Quantum Mechanical Current of the Pauli Equation

Abstract

We argue that the process of constructing the quantum mechanical current of the Pauli equation by copying the line of arguments used in the spin-0 case, i.e. the Schr\"{o}dinger equation, is ambiguous. We show that a non-relativistic reduction of the relativistic Dirac four-vector current is, however, capable of fully resolving the problem. This analysis reveals that the non-relativistic current of the Pauli equation should include an extra term of the form $\vec \nabla \vec \times (\psi^{\dagger} \vec \sigma \psi)$. We present an initial exploration of the potential consequences of this new 'spin-term' by solving the Pauli equation for crossed magnetic and electric fields and calculating the corresponding current.