One-electron probability densities and probability current densities
In order to derive the one-electron Pauli probability current density, we start by introducing the time-dependent Pauli equation in a slightly different form: \[\begin{equation} \require{physics} \pdv{t} \Psi(\vb{r}, t) = -i \thinspace \mathcal{H}^c(\phi, \vb{A}) \Psi(\vb{r}, t) \thinspace , \end{equation}\] where \(\mathcal{H}^c(\phi, \vb{A})\) is the one-electron Pauli Hamiltonian in the presence of the scalar and vector potentials \(\phi(\vb{r})\) and \(\vb{A}(\vb{r})\): \[\begin{equation} \mathcal{H}^c(\phi, \vb{A}) = \qty[ k^c(\vb{A}) - \phi(\vb{r}) ] \vb{I}_2 + \frac{1}{2} \boldsymbol{\sigma} \vdot \vb{B}(\vb{r}) \thinspace , \end{equation}\] in which \(k^c(\vb{A})\) is the scalar kinetic operator. The Pauli probability density \(\rho(\vb{r}, t)\) for one electron is: \[\begin{equation} \rho(\vb{r}, t) = \Psi^\dagger(\vb{r}, t) \Psi(\vb{r}, t) \thinspace , \end{equation}\] so in order to derive the associated probability current density \(\vb{j}(\vb{r}, t)\) which adheres to the probability continuity equation \[\begin{equation} \pdv{\rho(\vb{r}, t)}{t} + \boldsymbol{\nabla} \vdot{\vb{j}(\vb{r}, t)} = 0 \thinspace , \end{equation}\] we have to partially differentiate the Pauli distribution with respect to time.
In order to do so, we also require the Hermitian adjoint of the Pauli equation: \[\begin{equation} \pdv{t} \Psi^\dagger(\vb{r}, t) = i \thinspace \qty[ \mathcal{H}^c(\phi, \vb{A}) \Psi(\vb{r}, t) ]^\dagger \thinspace , \end{equation}\] where we will require the complex conjugate of the scalar kinetic operator: \[\begin{equation} k^{c, *}(\vb{A}) = T^c + i \vb{A}(\vb{r}) \vdot \grad + \frac{1}{2} \norm{ \vb{A}(\vb{r}) }^2 \thinspace . \end{equation}\] Working out the partial derivative and expanding the spinors in their two-component form, we can see that many of the terms cancel. As an intermediary result, we find: \[\begin{equation} \begin{split} \pdv{\rho(\vb{r}, t)}{t} = \sum_\sigma \bigg( &i \thinspace \qty[ T^c \Psi^*_\sigma(\vb{r}, t) ] \thinspace \Psi_\sigma(\vb{r}, t) -i \thinspace \Psi^*_\sigma(\vb{r}, t) \thinspace \qty[ T^c \Psi_\sigma(\vb{r}, t) ] \\ & - \vb{A}(\vb{r}, t) \vdot \qty[ \qty( \grad{ \Psi^*_\sigma(\vb{r}, t) } ) \thinspace \Psi_\sigma(\vb{r}, t) ] - \Psi^*_\sigma(\vb{r}, t) \thinspace \vb{A}(\vb{r}, t) \vdot \qty[ \grad{ \Psi_\sigma(\vb{r}, t) }] \bigg) \thinspace . \end{split} \end{equation}\] Trying to bring this expression into the following form of the continuity equation \[\begin{equation} \pdv{\rho(\vb{r}, t)}{t} = - \boldsymbol{\nabla} \vdot{\vb{j}(\vb{r}, t)} \thinspace , \end{equation}\] we should make sure a \(- \grad \vdot\) expression appears. Using the vector calculus property \[\begin{equation} \boldsymbol{\nabla} \vdot( \vb{A} f ) = \qty(\grad{f}) \vdot \vb{A} + f \thinspace \qty(\boldsymbol{\nabla} \vdot{\vb{A}}) \thinspace , \end{equation}\] and realizing that we are always working in Coulomb gauge, we find for the partial derivative of the Pauli probability density: \[\begin{equation} \begin{split} \pdv{\rho(\vb{r}, t)}{t} = - \grad \vdot \Bigg( \sum_\sigma \bigg[ & \frac{i}{2} \bigg( \Psi_\sigma(\vb{r}, t) \thinspace \qty[ \grad{ \Psi^*_\sigma(\vb{r}, t) } ] - \Psi^*_\sigma(\vb{r}, t) \thinspace \qty[ \grad{ \Psi_\sigma(\vb{r}, t) } ] \bigg) \\ &+ \Psi^*_\sigma(\vb{r}, t) \Psi_\sigma(\vb{r}, t) \thinspace \vb{A}(\vb{r}, t) \bigg] \bigg) \thinspace . \end{split} \end{equation}\] We see that we must then define the Pauli probability current density as follows: \[\begin{equation} \begin{split} \vb{j}(\vb{r}, t) = - \frac{i}{2} \sum_\sigma \bigg[ & \Psi^*_\sigma(\vb{r}, t) \thinspace \qty[ \grad{ \Psi_\sigma(\vb{r}, t) } ] - \qty[ \grad{ \Psi^*_\sigma(\vb{r}, t) } ] \thinspace \Psi_\sigma(\vb{r}, t) \\ &+ 2i \thinspace \Psi^*_\sigma(\vb{r}, t) \Psi_\sigma(\vb{r}, t) \thinspace \vb{A}(\vb{r}, t) \bigg] \thinspace . \end{split} \end{equation}\] Alternatively, we may write the Pauli probability current density as: \[\begin{equation} \vb{j}(\vb{r}, t) = -\frac{i}{2} \Bigg( \Psi^\dagger(\vb{r}, t) \thinspace \qty[ \grad{\Psi(\vb{r}, t)} ] - [ \grad{\Psi^\dagger(\vb{r}, t)} ] \thinspace \Psi(\vb{r}, t) % no qty + 2i \thinspace \Psi^\dagger(\vb{r}, t) \Psi(\vb{r}, t) \thinspace \vb{A}(\vb{r}, t) \Bigg) \thinspace , \end{equation}\] in two-component form, where the gradient of a 2-component spinor should be regarded as the \(2\)-vector of the gradient of the (scalar) components. Expressions for the Schrödinger current density are analogous. (Lazzeretti 2000)