Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)2470610.22363/2658-4670-2020-28-3-274-288ArticlesСтатьиResearch ArticleKuryshkin-Wodkiewicz quantum measurement model for alkaline metal atomsМодель квантовых измерений Курышкина-Вудкевича для атомов щелочных металловZorinAlexander V.ЗоринА. В.Candidate of Physical and Mathematical Sciences, assistant professor of Department of Applied Probability and Informaticszorin-av@rudn.ruPeoples’ Friendship University of Russia (RUDN University)Российский университет дружбы народов15122020283VOL 28, NO3 (2020)ТОМ 28, №3 (2020)27428828092020Copyright ©; 2020, Zorin A.V.Copyright ©; 2020, Зорин А.В.2020Zorin A.V.Зорин А.В.http://creativecommons.org/licenses/by/4.0https://journals.rudn.ru/miph/article/view/24706The 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.Конструктивная форма модели квантовых измерений Курышкина-Водкевича ранее была подробно разработана для квантовой задачи Кеплера. Для более сложных квантовых объектов такая конструкция неизвестна. В то же время стандартная (неконструктивная) модель квантовых измерений Холево-Хелстрома подходит для любого квантового объекта. В данной работе конструктивная модель квантовых измерений обобщена на более широкий класс квантовых объектов, то есть на оптический спектр атомов и ионов с одним валентным электроном. Анализ основан на экспериментальных данных об энергетическом упорядочении электронов в атоме по правилу Клечковского-Маделунга и на обосновании одночастичной потенциальной модели для описания энергетического спектра оптических электронов в атомах щелочных металлов. Представление возмущения одночастичного потенциала в виде свертки потенциала электрона в атоме водорода с функцией Вигнера некоторого эффективного состояния остова в представлении атома щелочного металла позволяет редуцировать все алгоритмы расчета для щелочных металлов к соответствующим алгоритмам для атома водорода.models of quantum measurementsenergy spectrum of alkali metal atomsmethod of single-particle potentialperturbation of discrete spectrum of an observableмодели квантовых измеренийэнергетический спектр атомов щелочных металловметод одночастичного потенциаламодель квантовых измеренийвозмущение дискретного спектра наблюдаемой1.A. V. Zorin and L. A. Sevastianov, “Hydrogen-like atom with nonnegative quantum distribution function,” Physics of Atomic Nuclei, no. 70, pp. 792-799, 2007. DOI: 10.1134/S1063778807040229.2.L. Sevastyanov, A. Zorin, and A. Gorbachev, “Pseudo-Differential Operators in an Operational Model of the Quantum Measurement of Observables,” in Mathematical Modeling and Computational Science, G. Adam, J. Buša, and M. Hnatič, Eds., vol. 7125, Berlin, Heidelberg: Springer Berlin Heidelberg, 2012, pp. 174-181. DOI: 10.1007/978-3- 642-28212-6_17.3.G. M. D’Ariano, U. Leonhardt, and H. Paul, “Homodyne detection of the density matrix of the radiation field,” Phys. Rev. A, vol. 52, R1801-R1804, 1995. DOI: 10.1103/PhysRevA.52.R1801.4.G. M. D’Ariano, “Measuring Quantum States,” in Concepts and Advances in Quantum Optics and Spectroscopy of Solids, T. Hakioglu and A. S. Shumovsky, Eds., Amsterdam: Kluwer Acad. Publishers, 1997, pp. 175- 202.5.A. S. Holevo, Statistical Structure of Quantum Theory, ser. Lecture Notes in Physics Monographs. Berlin: Springer, 2001, vol. 67. DOI: 10.1007/3-540-44998-1.6.C. W. Helstrom, Quantum Detection and Estimation Theory. New York: Academic Press, 1976.7.G. Ludwig, Attempt of an axiomatic foundation of quantum mechanics and more general theories, II, ser. Commun. Math. Phys. 2001, vol. 4. DOI: 10.1007/BF01653647.8.E. B. Davies and J. T. Lewis, “An operational approach to quantum probability,” Communications in Mathematical Physics, vol. 17, no. 3, pp. 239-260, 1970.9.M. Ozawa, Quantum reality and measurement: A quantum logical approach, ser. Foundations of Physics. 2011, vol. 41, pp. 592-607.10.M. Ozawa and Y. Kitajima, Reconstructing Bohr’s Reply to EPR in Algebraic Quantum Theory, ser. Foundations of Physics. 2012, vol. 42, pp. 475-487. DOI: 10.1007/s10701-011-9615-7.11.A. V. Zorin, L. A. Sevastianov, and N. P. Tretyakov, “Computer modeling of hydrogen-like atoms in quantum mechanics with nonnegative distribution function,” Programming and Computer Software, vol. 33, no. 2, pp. 94-104, 2007. DOI: 10.1134/S0361768807020077.12.V. A. Fock, Foundations of quantum mechanics. Mir Publishers, 1978.13.V. Kondratyev, The Structure of Atoms and Molecules. Univ Pr. of the Pacific, 2002.14.V. N. Ostrovsky, “What and How Physics Contributes to Understanding the Periodic Law,” Foundations of Chemistry, no. 3, pp. 145-181, 2001. DOI: 10.1023/A:1011476405933.15.V. M. Klechkovskii, The Distribution of Atomic Electrons and the Rule of Successive Filling of (n + l)-Groups [Raspredelenie atomnyh elektronov i pravilo posledovatel’nogo zapolneniya (n + l)-grupp]. Moscow: Atomizdat, 1968, in Russian.16.E. Madelung, Die Mathematischen Hilfsmittel des Physikers, 3rd edition. Berlin: Springer, 1936. DOI: 10.1007/978-3-662-21800-6.17.Y. N. Demkov and V. N. Ostrovskii, “Internal Symmetry of the Maxwell “Fish-eye” Problem and the Fock Group for the Hydrogen Atom,” JETP, vol. 33, no. 6, pp. 1083-1087, 1971.18.Y. N. Demkov and V. N. Ostrovsky, “n+l Filling Rule in the Periodic System and Focusing Potentials,” JETP, vol. 35, no. 1, pp. 66-69, 1972.19.V. A. Fock, “Hydrogen atom and non-Euclidean geometry,” Zs. Phys., vol. 98, p. 145, 1935.20.Y. Kitagawara and A. O. Barut, “Period doubling in the n+l filling rule and dynamical symmetry of the Demkov-Ostrovsky atomic model,” Journal of Physics B: Atomic and Molecular Physics, vol. 16, no. 18, pp. 3305-3327, 1983. DOI: 10.1088/0022-3700/16/18/006.21.Y. Kitagawara and A. O. Barut, “On the dynamical symmetry of the periodic table. II. Modified Demkov-Ostrovsky atomic model,” Journal of Physics B: Atomic and Molecular Physics, vol. 17, no. 21, pp. 4251- 4259, 1984. DOI: 10.1088/0022-3700/17/21/013.22.A. L. Kholodenko. (2020). “From Mendeleev to Seiberg-Witten via Madelung. Available from.” accessed Jul 16 2020, [Online]. Available: https://www.researchgate.net/publication/341597880.23.A. L. Kholodenko and L. H. Kauffman, “How the modified Bertrand theorem explains regularities of the periodic table I. From conformal invariance to Hopf mapping,” 2019. arXiv: 1906.05278.24.Y. B. Rumer and A. I. Fet, “The group Spin (4) the Mendeleev system,” Theor. Math. Phys, vol. 9, pp. 1081-1085, 1971. DOI: 10.1007/ BF01036944.25.V. V. Varlamov, “Group Theoretical Description of Periodic System of Elements [Teoretiko-gruppovoe opisanie periodicheskoj sistemy elementov],” Mathematical Structures and Modelling, vol. 46, no. 2, pp. 5-23, 2018, in Russian. DOI: 10.25513/2222-8772.2018.2.5-23.26.D. Kirzhnitz, Y. Lozovik, and G. Shpatkovskaya, “Statistical model of matter,” Sov. Phys. Uspekhi., vol. 18, no. 9, pp. 649-672, 1975. DOI: 10.1070/PU1975v018n09ABEH005199.27.A. Fet, Group Theory of Chemical Elements. Berlin: de Gryuter, 2016.28.L. Sevastianov, A. Zorin, and A. Gorbachev, “A Quantum Measurements Model of Hydrogen-Like Atoms in Maple,” in Computer Algebra in Scientific Computing, V. P. Gerdt, W. Koepf, E. W. Mayr, and E. V. Vorozhtsov, Eds., vol. 8136, Cham: Springer International Publishing, 2013, pp. 369-380. DOI: 10.1007/978-3-319-02297-0_30.29.B. Simon, “Tosio Kato’s work on non-relativistic quantum mechanics: Part 1,” Bulletin of Mathematical Sciences, no. 8, pp. 121-232, 2018. DOI: 10.1007/s13373-018-0118-0.30.B. Simon, “Tosio Kato’s work on non-relativistic quantum mechanics, Part 2,” Bulletin of Mathematical Sciences, vol. 9, no. 1, p. 1 950 005, 2019. DOI: 10.1142/S166436071950005X.31.A. V. Zorin, “Approximate Computation of States with Minimal Dispersion in Kuryshkin-Wodkiewicz Quantum Mechanics,” in 2019 11th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), Dublin, Ireland, 2019, 2019, pp. 1-5. DOI: 10.1109/ICUMT48472.2019.8971007.32.(2020). “Physical reference data.” accessed Jul 16 2020, [Online]. Available: https://www.nist.gov/pml/productsservices/physical-reference-data/.