Fock similarity transformation of the electronic Hamiltonian

The covariance of the one-electron Schrödinger equation in a gauge transformation of electromagnetic potentials is customarily discussed, introducing the relationships that describe their changes, induced by a generating function of position coordinates and time, together with a phase transformation of the wavefunction, according to a proposal by Fock, i.e.,² relying on three distinct hypotheses. The present note shows that (i) a single similarity transformation employing the Fock phase factor is sufficient to prove the covariance of the wave equation and (ii) the Fock transformations of the nonperturbed Hamiltonian ĥ⁽⁰⁾ and of the time-derivative operator determine the changes appearing in first and second-order Hamiltonians, ĥ⁽¹⁾ and ĥ⁽²⁾, a single term being obtained by Fock-transforming ĥ⁽¹⁾.