Unusual commutation relations in physics

Abstract

The angular momentum operator L̂ of a particle obeys the usual commutation rule L̂×L̂=iℏL̂. In contrast, the total angular momentum operator Ĵ of a set of particles relative to moving axes (as, for instance, when mounted on a tumbling molecule) obeys the anomalous commutation rule Ĵ×Ĵ=−iℏĴ. We give a pedagogical and mathematically nonsophisticated description of intermediate cases for which partial angular momentum operators, relative to various moving frames, obey unusual commutation relations that are neither usual nor anomalous. Insight into the origin of these unusual commutation relations provides a means of guessing the type of commutation rule that will be obeyed. In particular, it gives a way to avoid, as far as possible, the unusual commutation relations, which lead to complicated and nonsystematic expressions. Some examples from molecular physics are presented.