arXiv:2406.06989 · Journal of Mathematical Analysis and Applications

Regularized quantum motion in a bounded set: Hilbertian aspects

Fabio Bagarello, Jean‐Pierre Gazeau, Camillo Trapanı

Published Jun 15, 2024• 1 citation

Abstract

It is known that the momentum operator canonically conjugated to the position operator for a particle moving in some bounded interval of the line (with Dirichlet boundary conditions) is not essentially self-adjoint: it has a continuous set of self-adjoint extensions. We prove that essential self-adjointness can be recovered by symmetrically weighting the momentum operator with a positive bounded function approximating the indicator function of the considered interval. This weighted momentum operator is consistently obtained from a similarly weighted classical momentum through the so-called Weyl-Heisenberg covariant integral quantization of functions or distributions.

DOI: 10.1016/j.jmaa.2024.128631

Related papers

Canonical quantization on the half-line and in an interval based upon an alternative concept for the momentum in a space with boundaries

M.H. Al-Hashimi, U.‐J. Wiese

For a particle moving on a half-line or in an interval the operator pˆ = −i∂x is not self-adjoint and thus does not qualify as the physical momentum. Consequently canonical quantization based on pˆ fails. Based upon an alternative concept for a self-adjoint momentum operator pˆR, we show that canonical quantization can indeed be implemented on the half-line and on an interval. Both the Hamiltonian Hˆ and the momentum operator pˆR are endowed with self-adjoint extension parameters that characterize the corresponding domains D(Hˆ ) and D(pˆR ) in the Hilbert space. When one replaces Poisson brackets by commutators, one obtains meaningful results only if the corresponding operator domains are properly taken into account. The alternative concept for the momentum is used to describe the results of momentum measurements of a quantum mechanical particle that is reflected at impenetrable boundaries, either at the end of the half-line or at the two ends of an interval.

Jan 1, 202113 citations

Canonical quantization on the half-line and in an interval based upon an alternative concept for the momentum in a space with boundaries

M. H. Al-Hashimi, U.‐J. Wiese

For a particle moving on a half-line or in an interval the operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}=\ensuremath{-}i{\ensuremath{\partial}}_{x}$ is not self-adjoint and thus does not qualify as the physical momentum. Consequently canonical quantization based on $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}$ fails. Based upon an alternative concept for a self-adjoint momentum operator ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{R}$, we show that canonical quantization can indeed be implemented on the half-line and on an interval. Both the Hamiltonian $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}$ and the momentum operator ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{R}$ are endowed with self-adjoint extension parameters that characterize the corresponding domains $D(\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H})$ and $D({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}}_{R})$ in the Hilbert space. When one replaces Poisson brackets by commutators, one obtains meaningful results only if the corresponding operator domains are properly taken into account. The alternative concept for the momentum is used to describe the results of momentum measurements of a quantum mechanical particle that is reflected at impenetrable boundaries, either at the end of the half-line or at the two ends of an interval.

arXiv:2103.01715Physical Review ResearchJul 21, 202114 citations

Quantum smooth boundary forces from constrained geometries

Jean‐Pierre Gazeau, T Koide, D Noguera

Abstract We implement the so-called Weyl–Heisenberg covariant integral quantization in the case of a classical system constrained by a bounded or semi-bounded geometry. The procedure, which is free of the ordering problem of operators, is illustrated with the basic example of the one-dimensional motion of a free particle in an interval, and yields a fuzzy boundary, a position-dependent mass, and an extra potential on the quantum level. The consistency of our quantization is discussed by analyzing the semi-classical phase space portrait of the derived quantum dynamics, which is obtained as a regularization of its original classical counterpart.

arXiv:1902.07305Journal of Physics A Mathematical and TheoreticalSep 24, 201910 citations

On Quantum Dynamics on C*-Algebras

Igor Vasil'evich Volovich, Vsevolod Zhanovich Sakbaev

We consider the problem of constructing quantum dynamics for symmetric Hamiltonian operators that have no self-adjoint extensions. For an earlier studied model, it was found that an elliptic self-adjoint regularization of a symmetric Hamiltonian operator allows one to construct quantum dynamics for vector states on certain C *-subalgebras of the algebra of bounded operators in a Hilbert space. In the present study, we prove that one can extend the dynamics to arbitrary states on these C *-subalgebras while preserving the continuity and convexity. We show that the obtained extension of the dynamics of the set of states on C *-subalgebras is the limit of a sequence of regularized dynamics under removal of the elliptic regularization. We also analyze the properties of the limit dynamics of the set of states on the C *-subalgebras.

Proceedings of the Steklov Institute of MathematicsMay 1, 201818 citations

On Distributions of Self-Adjoint Extensions of Symmetric Operators

Franco Fagnola, Zheng Li

In quantum probability a self-adjoint operator on a Hilbert space determines a real random variable and one can define a probability distribution with respect to a given state.In this paper we consider self-adjoint extensions of certain symmetric operators, such as momentum and Hamiltonian operators, with various boundary conditions, explicitly compute their probability distributions in some state and study dependence of these probability distributions on boundary conditions.

Journal of Stochastic AnalysisJun 3, 20210 citations

Weyl quantization

Giampiero V. M. Esposito, Giuseppe Marmo, George Sudarshan

The commutator of position and momentum operators is first considered in the coordinate and momentum representations, and then in an abstract Hilbert space. The reader is then introduced to canonical operators with the associated canonical quantization of commutation rules, which consists of maps defined on a symplectic space and taking values in the set of unitary operators on a Hilbert space. If the symplectic space is finite-dimensional, all of its irreducible representations are unitarily equivalent. The Weyl exponentiated form of the commutation relations is described and investigated. One then arrives at the definition of Weyl systems, and it is shown how to recover from them the formulation in terms of self-adjoint operators which satisfy a generalized form of commutation relations. Further topics are the Heisenberg representation for temporal evolution and the generalized uncertainty relations. Unitary operators associated with symplectic linear transformations are then considered, and, within this framework, translations, rotations and the harmonic oscillator are considered. The Weyl programme is then reassessed. Lastly, the basic postulates of modern quantum theory are eventually studied: the probabilistic nature of the predictions of results of measurements; the correspondence between observables and self-adjoint operators on Hilbert spaces; the expected result of measuring an observable quantity; and the evolution of the state vector in systems which are not affected by any external influence. The chapter ends with an outline of rigged Hilbert space formalism and its applications. The commutator in wave mechanics In the first part of this book we studied wave mechanics within the framework of the Schrödinger equation.

Cambridge University Press eBooksMar 11, 20040 citations