Energy operator for non-relativistic and relativistic quantum mechanics revisited

TL;DR

This paper revisits the energy operator in quantum mechanics, demonstrating its gauge invariance and providing a first-principles derivation applicable to both non-relativistic and relativistic cases, aligning with recent proposals in time-independent fields.

Contribution

It offers a gauge-invariant method to define the energy operator from first principles, resolving ambiguities in both non-relativistic and relativistic quantum mechanics.

Findings

Energy operator is gauge invariant under transformations.

The method applies to stationary states in both regimes.

Results agree with recent proposals for time-independent fields.

Abstract

Hamiltonian operators are gauge dependent. For overcome this difficulty we reexamined the effect of a gauge transformation on Schr\"odinger and Dirac equations. We show that the gauge invariance of the operator $H-i\hbar\frac{\partial}{\partial t}$ provides a way to find the energy operator from first principles. In particular, when the system has stationary states the energy operator can be identified without ambiguities for non-relativistic and relativistic quantum mechanics. Finally, we examine other approaches finding that in the case in which the electromagnetic field is time independent, the energy operator obtained here is the same as one recently proposed by Chen et al. [1].