What is the limit $\hbar \to 0$ of quantum theory ?

Abstract

An analysis is made of the relation between quantum theory and classical mechanics, in the context of the limit $\hbar \to 0$. Several ways in which this limit may be performed are considered. It is shown that Schr\"odinger's equation for a single particle moving in an external potential $V$ does not, except in special cases, lead, in this limit, to Newton's equation of motion for the particle. This shows that classical mechanics cannot be regarded as emerging from quantum mechanics-at least in this sense-upon straightforward application of the limit $\hbar \to 0$.