Approximate time–energy uncertainty relationship from the fixed-energy sum over paths approach

In this article, we show how an approximate time–energy uncertainty relationship of the form ΔEΔt ≃ ћ can be derived in the context of the fixed-energy sum over paths approach to quantum bound systems. The relationship connects the indeterminacy Δt on the travel time of the quantum object to the width ΔE of the resonance in the approximate Green function corresponding to an allowed value of energy. The mathematical origin of the relationship is to be tracked to the Fourier transform relationship between the time propagator and energy-dependent Green function; however, the core of the derivation does not use advanced mathematics, may be carried out using mostly graphical and geometrical methods, and may provide insight to students on the meaning and origin of the time–energy uncertainty relationship. Our work may contribute to close a gap by which the time–energy relationship is most often taught with insufficient explanation at elementary level.