Abstract

We consider the first-order finite-difference expression of the commutator between d / dx and x. This is the appropriate setting in which to propose commutators and time operators for a quantum system with an arbitrary potential function and a discrete energy spectrum. The resulting commutators are identified as universal lowering and raising operators. We also find time operators which are finite-difference derivations with respect to the energy. The matrix elements of the commutator in the energy representation are analyzed, and we find consistency with the equality \. We apply the theory to the particle in an infinite well and for the Harmonic oscillator as examples.