Can a spin-half particle ever give more than two spots in a Stern-Gerlach experiment? -- the subtle physics of effective Hamiltonians

We show that a spin-1/2 particle can behave as if it were spin-$s$, and generate $2s+1$ spots in a Stern Gerlach measurement (albeit with a smaller gyromagnetic ratio). This arises from some subtle properties of effective Hamiltonians and Hamiltonians with constraints. Examples of implications of the effect in condensed matter are discussed. We also give some simple non-perturbative bounds for a system subjected to strong constraints.