Quantum Particle Dynamics in a Highly Singular 1D-Potential $U(x) = -\alpha \delta(x) + \beta \delta'(x)$ Superposed on a Well-Behaved One

TL;DR

This paper analytically derives the exact Green's function for a quantum particle in a 1D potential combining a delta and a delta prime function, revealing that electron transmission is blocked by the delta prime component.

Contribution

It provides a closed-form analytical solution for the Green's function of a 1D quantum system with a generalized singular potential and variable mass, advancing understanding of such singular interactions.

Findings

No electron transmission through the beta delta prime potential.

Exact Green's function constructed analytically.

Transmission unaffected by the well-behaved potential V(x).

Abstract

We examine the one-dimensional quantum dynamics of a Schroedinger particle in a potential represented by a generalized function of the form $U(x) = -\alpha \delta (x) + \beta d(\delta (x))/dx$ superposed on a well behaved potential $V(x)$. In this, we construct the full, exact Green's function for such a 1D system analytically in closed form, taking account of a spatially variable mass $m(x)$. Our result shows that there can be no electron transmission through the $\beta \delta '(x)$- potential, regardless of the presence of the $V(x)$- potential and $\alpha \delta (x)$, (with $\alpha \ne 0$).