On the 3n+l Quantum Number in the Cluster Problem

TL;DR

This paper explores a proposed quantum number 3n+l in sodium cluster spectra, comparing it to known quantum numbers in hydrogen and harmonic oscillator models to understand its significance in electronic shell structure.

Contribution

It investigates the nature of the quantum number 3n+l and its relation to established quantum numbers in fundamental quantum systems.

Findings

The quantum number 3n+l shows similarities to n+l and 2n+l in known quantum models.

The study clarifies the role of 3n+l in the electronic shell structure of sodium clusters.

Insights into the underlying quantum mechanics of cluster spectra are provided.

Abstract

It has recently been suggested that an exactly solvable problem characterized by a new quantum number may underlie the electronic shell structure observed in the mass spectra of medium-sized sodium clusters. We investigate whether the conjectured quantum number 3n+l bears a similarity to the quantum numbers n+l and 2n+l, which characterize the hydrogen problem and the isotropic harmonic oscillator in three dimensions.