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arXiv:nucl-th/0410018

The interparticle interaction and noncommutativity of conjugate operators in quantum mechanics. $H$-like atoms

A.I. Steshenko

Published Oct 5, 2004• 0 citations
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Abstract

A quantum mechanical model for the systems consisting of interacting bodies is considered. The model takes into account the noncommutativity of the space and impulse operators and the correlation equations for the indeterminacy of these quantities. The noncommutativity of the operators is here a result of the action of the interparticle forces and represents a natural generalization of the conventional commutation relation for the space and impulse operators for a single particle. The efficiency of the model is demonstrated by specific calculations concerning several well-known atomic systems.

DOI
10.48550/arxiv.nucl-th/0410018

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The interparticle interaction and noncommutativity of conjugate operators in quantum mechanics. HHH-like atoms
A quantum mechanical model for the systems consisting of interacting bodies is considered. The model takes into account the noncommutativity of the space and impulse operators and the correlation equations for the indeterminacy of these quantities. The noncommutativity of the operators is here a result of the action of the interparticle forces and represents a natural generalization of the conventional commutation relation for the space and impulse operators for a single particle. The efficiency of the model is demonstrated by specific calculations concerning several well-known atomic systems.
A quantum mechanical model for a system of interacting bodies, taking into account noncommutativity of the coordinate and impulse operators for different particles and also the correlation equalities, is considered. The noncommutativity of the operators is here the result of the action of interparticle forces and represents natural generalization of the conventional commutation relation between the coordinate and impulse operators for a single particle. The efficiency of the model is evidenced by the specific calculations of well-known systems of atomic physics.
A quantum-mechanical model for systems of interacting bodies that takes into account noncommutativity of the operators of coordinates and momenta of different particles and correlations of the uncertainties of these quantities is considered. Here, the noncommutativity of these operators is due to the effect of interparticle forces and arises as a natural generalization of the usual commutation relation between the coordinate and momentum operators for a single particle. The efficiency of the model is proven by specific calculations for systems known in atomic and nuclear physics.
We construct an effective commutative Schrödinger equation in Moyal space-time in (1+1)(1+1)(1+1)-dimension where both ttt and xxx are operator-valued and satisfy [t^,x^]=iθ\left[ \hat{t}, \hat{x} \right] = i θ[t^,x^]=iθ. Beginning with a time-reparametrised form of an action we identify the actions of various space-time coordinates and their conjugate momenta on quantum states, represented by Hilbert-Schmidt operators. Since time is also regarded as a configuration space variable, we show how an `induced' inner product can be extracted, so that an appropriate quantum mechanical interpretation is obtained. We then discuss several other applications of the formalism developed so far.
Some consequences of promoting the object of noncommutativity theta(ij) to an operator in Hilbert space are explored. Its canonical conjugate momentum is also introduced. Consequently, a consistent algebra involving the enlarged set of canonical operators is obtained, which permits us to construct theories that are dynamically invariant under the action of the rotation group. In this framework it is also possible to give dynamics to the noncommutativity operator sector, resulting in new features.
The thesis addresses the fundamental interplay between indistinguishability and interactions in few- and many-body quantum systems, treated in two separate approaches, both analytic in nature, using semiclassical methods. The few-body sector, considering a strictly fixed number of particles, is addressed within the framework of first quantization. By sticking to descriptions as (micro-)canonical ensembles the approach overcomes the inappropriate use of virial expansions. It is based on cluster expansions where quantum statistics are treated exactly and interparticle forces are described non-perturbatively. After a full formal development the cluster expansions are supplemented with short-time dynamical information that is equated with the description of smooth spectral properties or equivalently temperatures above strong quantum degeneracy, which, in the single-particle context, is the essential ingredient to what is known as semiclassical Weyl expansions. This reduction allows for the fully analytical description of universal features of large classes of systems by either truncating to a specific class of allowed collision processes, i.e., two-body collisions, or by extracting higher-order interaction kernels from related integrable models. The consistent combination with scaling considerations, minimal ground-state information and strong-coupling expansions leads to a toolbox providing most of the thermodynamic and spectral properties of one-dimensional systems with short-range interactions, all in terms of the lowest-order interaction kernels. The analytical results are in excellent qualitative and quantitative agreement with extensive numerical simulations for both integrable and non-integrable models of experimental relevance in cold atom physics. The many-body sector is in contrast treated in the second quantized formulation of quantum mechanics making use of the well-known collection of semiclassical methods that are subsumed under the name periodic orbit theory. As these methods apply in a highly system specific manner, the attractive Bose gas on a ring is chosen as exemplary model to show a case where the semiclassical quantization of mean-field dynamics, here considered as the classical analogue, can be used to obtain analytical non-trivial information on correlation-related many-body effects, especially related to quantum critical behavior in this model. After a simplification by truncating to the lowest single-particle momentum modes the consistent application of Einstein-Brillouin-Keller quantization yields a description valid for large numbers of bosons. Besides analytically reproducing numerical spectra to high accuracy, the approach enables one to extract asymptotic scaling laws for level spacings around quantum criticality, representing properties. These (finite) energy differences are intrinsically tied to inter-particle correlations and therefore elude a more conventional description by means of a mean-field theory extended with Bogoliubov quasi particles. An application to the scrambling time of the system, related to a pseudo quantum butterfly effect that stems from a local instability, underlines the importance of the method in the otherwise inaccessible regime of extreme particle numbers.