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Journal of Mathematical Physics

Confined quantum systems and their limits

F. Belgiorno, Franco Gallone

Published Feb 1, 2009• 3 citations

Abstract

In nonrelativistic quantum mechanics, the notion of the confined version of a system 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.

DOI: 10.1063/1.3054864

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In nonrelativistic quantum mechanics, the notion of the confined version of a system 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.

The Hilbert-Space Description By the static description of a physical system we mean the rules that assign specified mathematical objects to the states and to the physical quantities of the system, and the prescriptions for calculating the probability distribution of the possible values of every physical quantity when the state of the system is given. In the usual Hilbert-space formulation of quantum mechanics, to each physical system is attached a separable Hilbert space (generally infinite-dimensional) over the complex field. To every physical quantity is associated a linear, self-adjoint, not necessarily bounded operator on. If one deals with a strictly quantum system, then the converse is also generally assumed: every self-adjoint operator on represents some physical quan- tity. The restriction “strictly quantum” is necessary: if the system retains some nonquantum (i.e. classical) feature, so that it requires the algorithm of so-called “superselection rules” (see Chapter 5), then there are self-adjoint operators on that do not represent physical quantities. It should also be stressed that even in the strictly quantum case, most of the self-adjoint operators actually do not represent “interesting” physical quantities: only a few of them represent physical quantities that are useful and meaningful for the description of the physical system (e.g., energy, momentum, position, angular momentum). Therefore, by asserting that every self-adjoint operator on corresponds to some physical quantity we mean there is no a priori impossibility of devising such a correspondence, but we do not claim that this correspondence is present in the real work of experimental physics.

Based on the recent construction of a self-adjoint momentum operator for a particle confined in a one-dimensional interval, we extend the construction to arbitrarily shaped regions in any number of dimensions. Different components of the momentum vector do not commute with each other unless very special conditions are met. As such, momentum measurements should be considered one direction at a time. We also extend other results, such as the Ehrenfest theorem and the interpretation of the Heisenberg uncertainty relation to higher dimensions.

Based on the recent construction of a self-adjoint momentum operator for a particle confined in a one-dimensional interval, we extend the construction to arbitrarily shaped regions in any number of dimensions. Different components of the momentum vector do not commute with each other unless very special conditions are met. As such, momentum measurements should be considered one direction at a time. We also extend other results, such as the Ehrenfest theorem and the interpretation of the Heisenberg uncertainty relation to higher dimensions.

A reference frame F is described by the element g of the Poincaré group 𝒫 which connects F with a given fixed frame F0. If F is a quantum frame, defined by a physical object following the laws of quantum physics, the parameters of g have to be considered as quantum observables. However, these observables are not compatible and some of them, namely the coordinates of the origin of F, cannot be represented by self-adjoint operators. Both these difficulties can be overcome by considering a positive-operator-valued measure on 𝒫, covariant with respect to the left translations of the group, namely a covariance system. We develop a construction procedure for this kind of mathematical structure. The formalism is also used to discuss the quantum observables measured with respect to a quantum reference frame.

The purpose of the paper is to study the foundations of the main axioms of Quantum Mechanics. From a general study of the mathematical properties of the models used in Physics to represent systems, we prove that the states of a system can be represented in a Hilbert space, that a self-adjoint operator is associated to any observable, that the result of a measure must be the eigen value of the operator and appear with the usual probability. Furthermore an equivalent of the Wigner's theorem holds, which leads to the Schr{ö}dinger equation. These results are based on well known mathematics, and do not involve any specific hypothesis in Physics. They validate and explain the methods currently used, which are made simpler and safer, and open new developments. In the third edition of this paper developments have been added about the estimation of physical anomalies.