Energy operator for non-relativistic and relativistic quantum mechanics revisited

For quantum mechanics of a charged particle in a classical external electromagnetic eld the momen- tum and Hamiltonian operators are gauge dependent. For overcome this diculty we reexamined the eect of a gauge transformation on Schrodinger and Dirac equations. We show that the gauge invariance of the operator H i~ @ @t provides a way to nd the energy operator from rst principles. In particular, when the system has stationary states the energy operator can be identied without ambiguities for non-relativistic and relativistic quantum mechanics. Finally, we examine other approaches nding that in the case in which the electromagnetic eld is time independent, the energy operator obtained here is