Universal Raising and Lowering Operators for a Discrete Energy Spectrum

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 \([\hat{T},\hat{H}]=i\hbar \). We apply the theory to the particle in an infinite well and for the Harmonic oscillator as examples.