Quantum propagator and characteristic equation in the presence of a chain of $\delta$-potentials

TL;DR

This paper derives the quantum propagator and characteristic equation for systems with chains of delta potentials across various coordinate systems, demonstrating a simple method and applying it to confined oscillators with results matching numerical data.

Contribution

It introduces a straightforward and efficient method to obtain propagators and characteristic equations for delta potential chains in multiple coordinate systems, including applications to confined quantum oscillators.

Findings

Derived explicit propagators in rectangular, cylindrical, spherical coordinates.

Obtained characteristic equations for confined quantum harmonic oscillators.

Calculated energy eigenvalues approximately, matching existing numerical results.

Abstract

The quantum propagator and characteristic equation in the presence of a chain of $\delta$-potentials are obtained in the rectangular, cylindrical and spherical coordinate systems. The simplicity and efficiency of the method is illustrated via examples. As an application, the characteristic equation of a quantum harmonic oscillator confined to an infinite box is obtained. The roots of the characteristic equation, determining the energy eigenvalues of the restricted oscillator, are calculated approximately and compared with the existing numerical data.