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AIP conference proceedings

Coherent state quantization of time of arrival functions of confined and free particles

Daisy A. Romeo, Job A. Nable

Published Jan 1, 2022• 0 citations
Publisher

Abstract

This work presents coherent state quantization f↦∫f(x)|X〉〈X|dμ(X) applied to the classical time of arrival functions of the form T(q,p)=-μqp, expressed in different coordinate systems, corresponding to phase spaces of the p confined and free particle, respectively. The most important ingredient in this technique is the coherent state. Two ways of constructing coherent states will be adopted:via unitary group representations and via superposition method, applied to time of arrival function of one – dimensional confined and free particles.

DOI
10.1063/5.0092906

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This work presents coherent state quantization f↦∫f(x)|X〉〈X|dμ(X) applied to the classical time of arrival functions of the form T(q,p)=-μqp, expressed in different coordinate systems, corresponding to phase spaces of the p confined and free particle, respectively. The most important ingredient in this technique is the coherent state. Two ways of constructing coherent states will be adopted:via unitary group representations and via superposition method, applied to time of arrival function of one – dimensional confined and free particles.
The constructions of two classes of self-adjoint and compact first time of arrival operators for confined systems under arbitrary, everywhere continuous potential are detailed, extending in the interacting case the concept of confined quantum time of arrival operators first developed for the free particle. One class is the quantized confined time of arrival operators and another is the class of algebra preserving confined time of arrival operators. The former is the projection of the quantization of the classical time of arrival, and is constructed by solving the quantization problem of the multiple-valued and non-everywhere-real-valued classical time of arrival for arbitrary potential. The later arises to address the nonconjugacy with the Hamiltonian of the entire class of the quantized confined time of arrival operators, and is constructed by solving the obstruction to quantization present in Euclidean space. These two sets of operators coincide for linear systems; but differ for nonlinear systems, with the former as the leading term of the latter. The confined time of arrival operators for potentials representative of linear and nonlinear systems are numerically investigated and demonstrated to have the same dynamical behaviors as those of the free confined time of arrival operators. In particular, the eigenfunctions evolve according to Schrödinger equation such that the corresponding probabilities of locating the quantum particle in the neighborhood of the arrival point are maximum at their respective eigenvalues.
We construct the coherent states of a free particle by implementing a coordinate transformation in the extended configuration space, which establishes a correspondence between the solutions of the Schrödinger equation for a harmonic oscillator and those for a free particle. Using this framework, we derive analytical expressions for the coherent states of a free particle, avoiding the complexities of non-normalizable fiducial states, integrals of motion, or group-theoretic approaches. Our method provides a systematic way to characterize these states at any instant while ensuring that they satisfy the Robertson-Schrödinger uncertainty relation.
This work presents a generalized Weyl quantization for the time of arrival function for a system consisting of a free particle confined to move on a line segment.The generalized Weyl quantization relies crucially on the phase space of the classical system being quantized and on the choice of the group of symmetries of this space, which is non-unique.The Hilbert space of the corresponding quantum system is the representation Hilbert space of the unitary irreducible representations of the symmetry group.For the system considered here, the phase space of our time of arrival function is the cylinder × ℝ and the symmetry group (2) is the Euclidean motion group of the plane.We consider two coordinate systems leading to two distinct quantization.
Abstract Coherent states, in Wigner representation of quantum mechanics which is classical in structure with Wigner function playing the role of classical phase space probability distribution, and standard quantum mechanical expression for average values of corresponding Weyl quantum operators, general expression for matrix elements of the operator corresponding to phase of the oscillator by Weyl procedure, are obtained in the |n〉 basis. The general expression for matrix elements in the same basis of any other operator is also obtained. As the only mathematical technique necessary in the proposed procedure is a simple change of variables to polar coordinates in the corresponding integrals, this way to introduce Weyl phase operator, and some other operators in Weyl quantization, greatly simplifies necessary derivations and calculations.
Abstract This paper studies coherent quantization, the way operators in the quantum space of a coherent space—defined in the recent book ’Coherent Quantum Mechanics’ by the first author—can be studied in terms of objects defined directly on the coherent space. The results may be viewed as a generalization of geometric quantization, including the non-unitary case. Care has been taken to work with the weakest meaningful topology and to assume as little as possible about the spaces and groups involved. Unlike in geometric quantization, the groups are not assumed to be compact, locally compact, or finite-dimensional. This implies that the setting can be successfully applied to quantum field theory, where the groups involved satisfy none of these properties. The paper characterizes linear operators acting on the quantum space of a coherent space in terms of their coherent matrix elements. Coherent maps and associated symmetry groups for coherent spaces are introduced, and formulas are derived for the quantization of coherent maps. The importance of coherent maps for quantum mechanics is due to the fact that there is a quantization map that associates homomorphically with every coherent map a linear operator from the quantum space into itself. The quantization map generalizes the second quantization procedure for free classical fields to symmetry groups of general coherent spaces. Field quantization is obtained by specialization to Klauder spaces, whose quantum spaces are the bosonic Fock spaces. Implied by the new approach is a short, coordinate-free derivation of all basic properties of creation and annihilation operators in Fock spaces.