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Universal Raising and Lowering Operators for a Discrete Energy Spectrum

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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.

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  1. Physics Department, Cinvestav, Mexico City, Mexico

    Gabino Torres-Vega

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  1. Gabino Torres-Vega
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Correspondence to Gabino Torres-Vega.

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Torres-Vega, G. Universal Raising and Lowering Operators for a Discrete Energy Spectrum. Found Phys 46, 689–701 (2016). https://doi.org/10.1007/s10701-016-9997-7

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  • DOI: https://doi.org/10.1007/s10701-016-9997-7

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