We define the free energy functional in the Ginzburg-Landau theory as
$$ \mathcal{F} = F_n + \alpha |\psi|^2 + \frac{\beta}{2}|\psi|^4 + \frac{1}{2m^*}\big|(-i \hbar \nabla + e^*\mathbf{A})\psi \big |^2 + \frac{|\mathbf{B}|^2}{2 \mu_0}, $$
where:
- $\psi$ is a macroscopic wavefunction,
- $\alpha$ and $\beta$ are phenomenological parameters,
- $m^*$ is the effective mass of the charge carrier,
- $e^*$ is the effective charge of the charge carrier,
- $\mathbf{A}$ is a vector potential.
Moreover, we assume a static magnetic field, which leads to
$$ \nabla \times \mathbf{B} = \mu_0 \mathbf{j}. $$
In the book that I'm reading it's said, that the minimization of the free energy with respect to the vector potential
$$ \frac{d \mathcal{F}}{dA} = 0 $$
leads to the following equation for the dissipation-less electric current density
$$ \mathbf{j} = \frac{e^*}{2m^*} [\psi^*(-i \hbar \nabla - e^* \mathbf{A})\psi + c.c].$$
Could you explain please, how is it derived?