Derivation of the gauge-independent relation between the phase and the electric field

The effective wavefunction and the charge current are given as: --- (3.4) ---(3.13)The wavefunction follows the Schrödinger equation: --- (3.5)Without losing the generality, we can take the Coulomb gauge: --- (1)Now, we assume that the charge density is constant and uniform: In this case, the curre…