We present an alternative treatment for simple time-independent quantum systems in one dimension, which can be used in the context of an elementary introduction to quantum physics using the Feynman approach. The method is based on representation of the energy-dependent propagator (or Green function) as a sum of complex amplitudes over all possible paths, classical and non-classical, at fixed energy. We treat both confined and open systems with piecewise-constant potentials, obtaining exact results. We introduce an approximation scheme to extend the method to smooth potentials, recovering the Van Vleck-Gutzwiller propagator. Finally, we discuss the educational application of the method.

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To do so, it is not necessary to introduce the Schrödinger equation. The coefficients can be derived starting from the elementary propagator (12) with unknown Cν, and deriving them by imposing continuity and differentiability of the resulting Green function for the step potential.

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International GeoGebra Institute
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30.

Note that the expansion in Eq. (54) is valid for all discretized intervals except the one containing the classical turning point, for which necessarily ϵVx(i)>EV(i) and the first-order approximation is not meaningful. This justifies a separate treatment of tCT as in Eq. (49).

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