Simple models exhibiting spontaneous symmetry breakdown in classical and non-relativistic quantum mechanics

TL;DR

This paper analyzes classical and quantum models demonstrating spontaneous symmetry breakdown, introduces a simple quantum toy model exhibiting this phenomenon, and discusses pedagogical models' advantages and limitations.

Contribution

It presents a new simple quantum toy model showing spontaneous symmetry breaking and discusses the pedagogical value of existing models.

Findings

Classical model exhibits spontaneous symmetry breakdown.

Quantum toy model demonstrates internal symmetry breaking.

Discussion of pedagogical models' strengths and weaknesses.

Abstract

First, the properties of a classical model of spontaneous symmetry breakdown are analyzed. Then, the pros and cons of some pedagogical non-relativistic quantum-mechanical models, also used to illustrate spontaneous symmetry breakdown, are discussed. Finally, a simple quantum-mechanical toy model (a spinor on the line, with a spin-dependent interaction) is presented, that exhibits the spontaneous breaking of an internal symmetry.