Confined quantum systems and their limits

In nonrelativistic quantum mechanics, the notion of the confined version of a sys- tem is proposed and discussed. Confined observables can turn out to be represented by just positive operator valued measures also when their nonconfined equivalents are represented by self-adjoint operators, and this nearly always happens for confined linear momentum observables. However, the self-adjoint operators of the nonconfined system can still be used to compute expected results and variances of confined observables. As a result, the precision with which some confined observables can be measured has limits which ultimately derive, through the Heisenberg commutation rules, from the covariance properties of nonconfined kinematical observables. Two conditions are proposed to characterize the approximation of a nonconfined system by a sequence of confined ones. A nonconfined system is deemed to be approximated by a sequence of confined systems if all probabilities determined by any observable of the nonconfined system with respect to nonconfined states can be approximated by confined counterparts and if the time evolution of any state of the nonconfined system can be approximated in any finite time interval by the time evolution of confined states. A rather general example of this situation is discussed.