What is the limit ℏ → 0 of quantum theory ?

U. Klein*
University of Linz
Institute for Theoretical Physics
A-4040 Linz, Austria

August 24, 2012

*ulf.klein@jku.at

Abstract

An analysis is made of the relation between quantum theory and classical mechanics, in the context of the limit ℏ → 0 . Several ways in which this limit may be performed are considered. It is shown that Schrödinger’s equation for a single particle moving in an external potential V does not, except for special cases, lead, in this limit, to Newtons equation of motion for the particle. This shows that classical mechanics cannot be regarded as the limiting case of quantum mechanics for ℏ→ 0 .

1 Introduction
2 Two examples for classical limit relations
3 The ’standard limit’ of quantum theory
4 The ’deterministic limit’ of the ’standard limit’ of quantum theory
5 The ’deterministic limit’ of quantum theory
6 The ’combined limit’ of quantum theory
7 Can the ’combined limit’ be performed for all potentials ?
8 Discussion
9 Conclusion
10 Appendix
References