Dimensions, nodes and phases in quantum numbers

TL;DR

This paper offers a unified perspective on quantum numbers in quantum mechanics by connecting dimensions, boundary conditions, and inputs, clarifying their roles across different systems.

Contribution

It introduces a coherent framework that unites various ways quantum numbers are presented, highlighting underlying commonalities and sophisticated connections.

Findings

Provides a unified view of quantum numbers across systems

Clarifies the role of dimensions and boundary conditions

Establishes connections between different quantum number schemes

Abstract

Students of quantum mechanics encounter discrete quantum numbers in a somewhat incoherent and bewildering number of ways. For each physical system studied, quantum numbers seem to be introduced in its own specific way, some enumerating from 1 and others from 0, without a common uniting thread. This essay presents a point of view that builds on dimensions, boundary conditions and various inputs that, while known, are often not brought together to present a simple, consistent picture. At the same time, some surprisingly sophisticated connections are also made.