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Вестник Российского университета дружбы народов. Серия: Математика, информатика, физика

Kuryshkin-Wodkiewicz quantum measurement model for alkaline metal atoms

Alexander V. Zorin

Published Oct 6, 2020• 2 citations
Publisher

Abstract

The constructive form of the Kuryshkin-Wodkiewicz model of quantum measurements was earlier developed in detail for the quantum Kepler problem. For more complex quantum objects, such a construction is unknown. At the same time, the standard (non-constructive) model of Holevo-Helstrom quantum measurements is suitable for any quantum object. In this work, the constructive model of quantum measurements is generalized to a wider class of quantum objects, i.e., the optical spectrum of atoms and ions with one valence electron. The analysis is based on experimental data on the energy ordering of electrons in an atom according to the Klechkovsky-Madelung rule and on the substantiation of a single-particle potential model for describing the energy spectrum of optical electrons in alkali metal atoms. A representation of the perturbation of a single-particle potential in the form of a convolution of the potential of an electron in a hydrogen atom with the Wigner function of a certain effective state of the core in an alkali metal atom representation allows reducing all calculation algorithms for alkali metals to the corresponding algorithms for the hydrogen atom.

DOI
10.22363/2658-4670-2020-28-3-274-288

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The constructive form of the Kuryshkin-Wodkiewicz model of quantum measurements was earlier developed in detail for the quantum Kepler problem. For more complex quantum objects, such a construction is unknown. At the same time, the standard (non-constructive) model of Holevo-Helstrom quantum measurements is suitable for any quantum object. In this work, the constructive model of quantum measurements is generalized to a wider class of quantum objects, i.e., the optical spectrum of atoms and ions with one valence electron. The analysis is based on experimental data on the energy ordering of electrons in an atom according to the Klechkovsky-Madelung rule and on the substantiation of a single-particle potential model for describing the energy spectrum of optical electrons in alkali metal atoms. A representation of the perturbation of a single-particle potential in the form of a convolution of the potential of an electron in a hydrogen atom with the Wigner function of a certain effective state of the core in an alkali metal atom representation allows reducing all calculation algorithms for alkali metals to the corresponding algorithms for the hydrogen atom.
The structural form of the Kuryshkin-Wodkiewicz model of quantum measurement was developed in detail for quantum Kepler problem. For more complex objects such quantumstructure is unknown. At the same time, a standard (non-structural) model of quantum measurement proposed by Holevo-Helstrom is suitable for any quantum object. The aim of thiswork is to spread the structural model of quantum measurement to a broader class of quantum objects a model of quantum measurements of optical spectra of atoms and ions with one valence electron. In this work the Kuryshkin-Wodkiewicz model with implementation of Weyl-Kuryshkin quantization rule is applied to the extended quantum Kepler problem of quantum systems with one valence electron, such as alkali metal atoms. The proof of the consistency of the model is based on two Kato theorems about compact perturbations of operators. In the proof process the explicit form of the discrete spectrum of the valence electron for various spectral series was achieved with dependence on the serial parameters of the disturbance spectrum of an isolated object in the process of quantum measurement.
An algorithm for describing operator structures and calculating energy levels for hydrogen-like atoms in the quantum mechanics with nonnegative distribution function (Kuryshkin’s quantum mechanics) is developed. The algorithm is implemented in Maple. An original library of necessary functions and quantities (Coulomb wave functions, Sturm functions, their Fourier images and kernels of integral transformations, Clebsch-Gordan coefficients, products of spherical harmonics, etc.) has been created. In accordance with the quantization rules in Kuryshkin’s quantum mechanics, operators of observable quantities are computed. Based on the Hamiltonian obtained, energy levels of a hydrogen-like atom are calculated by the Ritz method (in the basis of Sturm functions). Numerical values of model parameters are determined by comparing results obtained with experimental data for hydrogen and alkali metals taken from the NIST Atomic Spectra Database Levels Data library.
The transfer of quantum information via quantum communication channels necessarily includes the process of quantum measurements of output. If we ignore the full description of the functioning of such channels, and focus only on the procedure of quantum measurements, we should work with the most mathematically rigorous standard Holevo model of quantum measurements. The Kuryshkin-Wodkiewicz quantum mechanics is a constructive version of the standard model, which includes a description of the phase representation of quantum mechanics. The longest living pure states in the dissipative quantum model under study are states with minimal dispersion. The work is devoted to an approximate numerical solution of the problem of finding such states.
Quantum theory of measurements is an extremely important part of quantum mechanics. Currently perturbations by quantum measurements of observable quantities of atomic systems are rarely taken into account in computing algorithms and calculations. In the previous studies of the authors, constructive model of quantum measurements has been developed and implemented in the form of symbolic and numerical calculations for the hydrogen-like atoms. This work describes a generalization of these results to the alkali metal atoms.
We report a theoretical calculation of the relativistic corrections to the binding energy of positronic alkali-metal atoms. The ground state for the positronic alkali-metal atom is a loosely bound state with the structure of an alkali-metal ion surrounded by a positronium cloud. The correlation between the valence electron and positron in the field of a residual core ion is taken into account using a three-body model. Relativistic corrections to the binding energy for positronic alkali-metal atoms are evaluated based on the Breit-Pauli perturbation theory up to the second order of the fine structure constant. The relativistic corrections caused by the interaction between the valence electron and core ion was evaluated from the decomposed expectation values of the perturbation Hamiltonian. We found that the importance of relativistic corrections to the binding energy, namely the ratio of the net relativistic correction in the total binding energy, is remarkably enhanced in the loosely bound states of positronic alkali-metal atoms, where the charge of the valence electron is screened by the positron as positronium in the outermost region, while the electron is released from the positron near the nucleus. The results imply that positronic atoms with a dominant positronium can serve as an apt testing ground for relativistic quantum mechanics.
For the last ten years, the production of polarised electrons has been of increasing interest in different fields of physics. One of the methods used to obtain polarised electron beams is based on the multiphoton ionisation of heavy alkali-metal atoms. The authors give a systematic analysis of the theoretical tools necessary for the study of the experimental quantities which can be measured in multiphoton ionisation experiments for moderate intensities in the non-resonant region. They use a description of the problem based on the density operator. The expansion of this density operator over the complete set of irreducible spherical tensor operators enables them to describe the different parameters attainable by experiments and so to construct the density matrix. This formalism allows the authors to study the total transition probabilities, the angular distributions and the differential polarisation degrees along any direction. They use the quantum defect theory as an atomic model. The atomic core polarisation is fully taken into account. The infinite summations running over the whole unperturbed atomic spectrum (discrete plus continuum) related to time dependent perturbation theory are performed using the Green's function formalism.