Quantum Mechanics of Multi-Prong Potentials

TL;DR

This paper investigates the quantum behavior of particles in multi-prong potentials, revealing unique energy level patterns, applying supersymmetric methods, and analyzing scattering and tunneling phenomena for arbitrary prong configurations.

Contribution

It introduces a comprehensive analysis of N-prong quantum potentials, including solutions for arbitrary lengths and the application of supersymmetric quantum mechanics techniques.

Findings

Energy levels show alternating degeneracy patterns for identical prongs.

Supersymmetric quantum mechanics can generate new solutions for these potentials.

Backscattering increases with the number of prongs in scattering scenarios.

Abstract

We describe the bound state and scattering properties of a quantum mechanical particle in a scalar $N$-prong potential. Such a study is of special interest since these situations are intermediate between one and two dimensions. The energy levels for the special case of $N$ identical prongs exhibit an alternating pattern of non-degeneracy and $(N-1)$ fold degeneracy. It is shown that the techniques of supersymmetric quantum mechanics can be used to generate new solutions. Solutions for prongs of arbitrary lengths are developed. Discussions of tunneling in $N$-well potentials and of scattering for piecewise constant potentials are given. Since our treatment is for general values of $N$, the results can be studied in the large $N$ limit. A somewhat surprising result is that a free particle incident on an $N$-prong vertex undergoes continuously increased backscattering as the number of prongs is increased.