Simple models exhibiting spontaneous symmetry breakdown in classical and   non-relativistic quantum mechanics
  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. 
R. Munoz, A. Garcia-Quiroz, Ernesto Lopez-Chavez and Encarnacion   Salinas-Hernandez
math-ph hep-th math.MP