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UDC 519.62; 530.145; 519.614 Kuryshkin-Wodkiewicz Model of Quantum Measurements for Atoms and Ions with One Valence Electron
A. V. Gorbachev, L. A. Sevastianov, A. V. Zorin
Peoples' Friendship University of Russia, Moscow, Russia
The structural form of the Kuryshkin-Wodkiewicz model of quantum measurement was developed in detail for quantum Kepler problem. For more complex objects such quantum structure 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 this work 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.
Key words and phrases: quantum measurement models, quantization rule, a relatively compact perturbation of the observable operator, perturbation of the observable discrete spectrum
1. Introduction
The energy spectrum En = — of the valence electron in a hydrogen atom is
described by a discrete spectrum of the Hamiltonian operator H = — y — 1 of the quantum
22
Kepler problem with the Hamiltonian function H (q,p) = ^ — . The measured
spectrum of the valence electron, except from the operator, is also dependent on the quantum state p of the probe of the measuring instrument, i.e. It is described by a discrete spectrum of the measured observable Op (H) = Ow (H * Wp.) [1,2].
The constructive form of the Kuryshkin-Wodkiewicz model of quantum measurement is developed in detail for quantum Kepler problem [3] and quantum oscillator [4,5]. For more complex objects such quantum construction is unknown. At the same time, a standard (non-constructive) model of quantum measurements of Holevo-Helstrom is suitable for any quantum object, any quantum system [6,7].
The aim of this work is to spread the constructive model (of quantum measurements) of Kuryshkin-Wodkiewicz to a wider class of quantum objects, to construct a model of quantum measurements of optical spectra of atoms and ions with one valence electron. Hydrogen-like atoms - a hydrogen atom, muonic atoms, ions of different charges with a single electron. We are interested in the atoms of the alkali metals, consisting of a core (a nucleus together with all the filled electron shells) and the valence electron and ions with one valence electron.
2. Hydrogen-Like Atoms
Theory of the hydrogen atom in quantum mechanics is a theory of an electron in a hydrogen atom. Its energy spectrum under theoretical analysis where electron is treated
Received 24th March, 2016.
as an isolated quantum object has a very simple form
R
E" = - (D
The measurement process violates the isolated state of the quantum object, transferring it to an open state, forming a part of a complex system: "object+probe" [8-11].
The measured energy spectrum of an electron in a hydrogen atom is perturbed with respect to the spectrum (1):
En = En + 5En (2)
There is a description problem (of constructing a mathematical model) of the measured values of the optical spectrum of the hydrogen atom. This model is a Weyl-Kuryshkin quantization rule equipped with: a mixed quantum state {pk} of the probe, smoothed (perturbed) classic observed A * W{Vk} (q,p), applied to it the Weil quantization rule 0{Vk} (A) = Ow (A). Next step will be a theoretical study of the spectrum of this operator and the numerical calculation of parts of the discrete spectrum affiliated with {(pk} [12-15].
First remark. While the experimental data is obtained with respect to a valence electron spin, the model not considers the spin state. As a result, at the current stage the studied model can not fully describe quantum interaction. However, on the one hand, it is adequate for the averaged over spin experimental values, on the other hand a model can be generalized taking into account the electron spin by Stratonovich-Weyl quantization [16-18].
For the hydrogen atom model is verified with relative accuracy ~ 10-16 (there is no special achievement in it). Before discussing the dependence of the perturbations 5En of the hydrogen atom under the influence of the measuring device with a quantum probe in the state {ifik} let us recall what is known about the discrete energy spectrum of the valence electron in an isolated atom of an alkali metal with a serial number Z.
3. The Energy Spectrum of the Valence Electrons in the Atoms
of the Alkali Metals
For atoms with more than one electron, even for the most simple, Schrodinger equation can not be solved immediately neither with analytical nor numerical methods. For this reason, the study of the spectra of many-electron atoms is based on an approximate model. The most appropriate approach turned out to be the one [19] that maintains an idea of the individual state of the electron in an atom, and the whole state of the atom is determined by a set of states of the electrons with respect to their interaction. This approach helps us obtain some general information about the system's possible energy levels for a given atom and their mutual arrangement.
General description of states of electrons in the atom is based on the assumption that each electron moves in an effective centrally symmetric field produced by the nucleus and all the other electrons. More detailed analysis focuses on the non-central part of the electrostatic and magnetic interaction of electrons. In the theory of atomic spectra these interactions are usually described as small corrections to the centrally symmetric field using methods of perturbation theory. Perturbation does not change the number of possible states of the system [19]. The calculation of the energy parameters can find practical use in various aspects of the atomic spectra theory. The key value of this calculations is determination of the wave functions. They are used in the calculation of the probabilities of radiative transitions, the effective excitation cross-sections and other characteristics of the atom. This is the main problem of the calculation of many-electron atoms, as energy levels can be obtained (and with high accuracy) from experiments.
In the approximation of a complete separation of the electronic variable the probability of radiative transitions k ^ k' between states k and k' can be expressed [19] through one-electron radial integrals
R,rJ = J PK (r) rPK' (r) dr. (3)
Therefore, the main task in the calculation of the transition probabilities is to find radial functions PK (r), PK' (r). For all of the atoms and ions, with the exception of those with a single electron, radial functions can be found only by approximate methods. Main methods of approximate calculation of the radial functions are different versions of variational methods and semi-empirical methods. All semi-empirical methods use of experimental values of energy levels.
Variational methods provide good approximation quality of functions PK (r) in the area of values r, which is the most significant in the calculation of energy. For large values of r these functions can be very inaccurate. With the help of semi-empirical methods it is easier to get functions P7 (r), exact at large r, i.e. just in the region which is most important for the calculation of transition probabilities. Therefore, much more simple semi-empirical method gives a better agreement with experiment (the accuracy of calculation of the transition probabilities) than the method of self-consistent field [19].
In the method of self-consistent field wave functions are calculated simultaneously with eigenvalues of the system of differential equations, with energy parameters EK. Different approach is more preferable for calculating the radiation transition probabilities. We can pre-define values EK and lookup such single-electron radial functions PK (r) that calculated values EK coincided with pre-defined. In this case the problem of self-consistent defined by the system of equations field is usually replaced by a single equation for the optical electron in an effective field. This equation has the form
№ — t + ^ + ™ — ^> = 0. (4)
The energy parameter EK is equal to the difference between the atom energy Ea and the "frozen" atomic core energy Ei [19]. Accuracy of functions PK (r) largely depends on how close the selected value EK is to the true value of the difference Ea — Ei. In the semi-empirical method, the energy parameter EK is equal to the experimental value of the ionization potential IK. Even for the alkaline earth atoms, and even more so for the alkali, it is the most appropriate to apply the semi-empirical method of calculations [19]. When choosing an effective potential VK (r) different approaches are available. However, in all cases, functions PK (r) have good asymptotic because the behavior of these functions for large r is determined by pre-selection of EK.
One of approaches is to choose potential VK (r) as a function of some fitted parameter whose value satisfies both boundary conditions. (In general case, the selected value E7 does not belong to eigenvalue of the equation (4), an thus it does not necessary have to satisfy both of the boundary conditions: P7 (0) = 0, P7 (r)-> 0).
4. Generalization of Kuryshkin-Wodkiewicz Model on the Valence Electrons of Alkali Metal Atoms
The spectral composition of terms (excluding spin interaction, relativistic corrections, etc.) of an isolated hydrogen atom can be easily calculated:
E„ = - 2-2. (5)
Upon splitting of the terms, they acquire corrections 5Eni so that
Eni = - 2^ + $Enl. (6)
In the measurement process (due to "interaction" with the quantum probe) the measured energy spectrum (after the projection on the original Hilbert space) becomes
R
Eni,K = - 2^ + $Eni + 5EK, (7)
where k is the multi-index of the quantum probe state. This is a standard approach for the hydrogen atom.
The spectral composition of terms of "hydrogen-like" alkali metals atoms is defined by the equation
Eni =--;-. (8)
2( n + a?)
Upon splitting of the terms, they acquire corrections
R
Enl,mms = + 3Eni,mms. (9)
, 2( n +
In the process of measuring the "measured" energy spectrum takes the form
R
Enl,mms,K = ; ^ + $Eni,mmB + ÔEK. (10)
2( n + af)
There is no need to take into account the additional contributions of fine structure and hyperfine structure in the formula (6) and (9) of the measured values of observables -energy before generalization of the Weil-Kuryshkin quantization rules to the quantization rule of Kuryshkin-Stratonovich. Consequently, the measured values of the terms should be limited by the formulas
R
= - + 6E1 (11)
and
E-i,K = ---+ SE-. (12)
' 2( n + af)
Moreover, contributions SE1 and ÔE'ff are obtained from the convolution V (r) * wivk} (q,P), where
y1 (r) = -1 (13)
and
1 tyZ
V- (r) = -- + — (14)
ry> iytZi
As a result the quantization rule of Weil-Kuryshkin is adequate for the alkali metals to the same extent as the potential VK (r, $), with respect to the small corrections to the centrally symmetric field of core.
5. The Model of Quantum Measurements of Kuryshkin-Wodkiewicz for Atoms and Ions of Alkali Metals
Recall the form
Op (H) = H + ^a^^ + Vj (r, cos 0})
of the Hamiltonian of hydrogen atom perturbed by the probe measurement in a quantum state
P = J2ai № (bj}) (bj}|.
Therefore, perturbation of the potential V1 (r) measured by probe in the state p has a form 5 Vp = ajVj (bj ,r, cos0) that is
Op( - J) = Ow( (-^ * W = -1 + SVP (r) .
Ritz method makes it possible with the help of 5VP (f) to calculate 5pEK for the
first spectral lines 5P\^ = (APK — A*) through the calculations of eigenvalues of the Ritz matrix of dimension N. In the works [20-23] it is shown how to restore the state p by disturbances 5PEK. Thus, on the perturbation 5VP (f) one can calculate
p = Y1 aj \Pj (bj)) {pj (bj)| such that j
(—1) *WP (r,p) = — 1 + SVP (P) .
As a result, we get
Theorem 1. For any mixed state p = J2aj \pj (bj)) {pj (bj)\ of a quantum probe
j
there exists a unique perturbation 5Vp (f) of potential V1 (f) = — 1 being V1 (r)-compact and equal to zero at infinity. And vice versa, to any V1 (r)-compact and equal to zero at infinity disturbance 5Vp (f) of potential V1 (r) corresponds the state p = Y1 cj \Pj) {Pj \
.space L2
P3N
uniquely defined in a subspace HN = < cj№j( of minimal dimension N in the Hilbert
Let us use the result of Kato's theorem [24]: If the operator O (H) = H + V (f) of the potential V can be written as the sum of two functions, one of which is continuous and bounded, and the other is the square-integrable on R3, while V (f) ——-> 0, then
| v\ —y^o
V is ii-compact. Any perturbation VK (r, cost?) of the centrally symmetric field Vz (r) in (14) in the problem (4) decreases at infinity, and thus according to Kato theorem Vz + VK is ii-compact, vanishing at infinity. According to the Statement 1 for (p) = yz (r) + VK (r, cost?) there is a state pz = Cj \Pj) {Pj \ defined uniquely in a subspace ( N ]
HN = { 12 cjPj\ of minimal dimension N in the Hilbert space L2 (R3) such that
j=1
(V1 *Wpz) (r, p) = (r). Thus, for the measured Hamilton operator of a valence electron in the alkali metal atom the measured potential is:
v£p(p) = c^f * wp) (r,p) = {(V1 * wpz) * Wp} (r,p).
Hence, is valid
Theorem 2. The quantization rule of Kuryshkin-Wodkiewicz applied to the valence electron in the alkali metal atom with an approximate pseudopotential (p) been measured by the quantum probe in the state p = Y1 cj Vh) {%h I take the form of quantization rule of Kuryshkin-Weyl with the potential (V^ * Wp) (r).
Corollary 1. Quantum measurement by the quantum probe in the state p of the energy spectrum of the valence electron in the alkali metal atom with state values
t^Z _
— —
R
nl , „x2
2(n + aZ)2
of (8), will result in measured value
Ezl'p — —R—Z^ + ÖEz
2(n + *Z)
of (12).
6. Conclusion
In [2,3,25,26] of quantum measurements applied to the quantum problem of Kepler has been implemented. In previous works a particular modification the model of quantum measurement has been applied to the problem of quantum oscillator [4,5]. In this paper the Kuryshkin-Wodkiewicz model with implementation of Weyl-Kuryshkin quantization rule [3,25,26] is applied to the quantum systems with one valence electron like alkali metal atoms Kuryshkin-Wodkiewicz model implementing quantization rule Weil-Kuryshkin, extended to the quantum system with one valence electron, such as alkali metal atoms. The proof of the consistency of the model is based on two Kato theorems [24]. The explicit form of the discrete spectrum of the valence electron for various spectral series, depending on the serial parameters of the disturbance spectrum of an isolated object in the process of quantum measurement was achieved.
References
1. A. V. Zorin, L. A. Sevastianov, Hydrogen-Like Atom with Nonnegative Quantum Distribution Function, Physics of Atomic Nuclei 70 (2007) 792-799.
2. A. V. Zorin, L. A. Sevastianov, Kuryshkin-Wodkiewicz Quantum Measurement Model, Bulletin of PFUR. Series "Mathematics. Information Sciences. Physics" 3 (2010) 98-103, in Russian.
3. L. A. Sevastianov, A. V. Zorin, A. V. Gorbachev, Pseudo-Differential Operators in the Operational Model of a Quantum Measurement of Observables, LNCS 7125 (2012) 174-181.
4. G. M. D'ariano, U. Leonhardt, H. Paul, Homodyne Detection of the Density Matrix of the Radiation Field, Physical Review A 52 (3) (1995) R1801.
5. G. M. D'ariano, Measuring Quantum States, in: Quantum Optics and the Spectroscopy of Solids, Springer, 1997, pp. 175-202.
6. A. S. Holevo, Statistical Structure of Quantum Theory, Vol. 67, Springer Science & Business Media, 2003.
7. C. W. Helstrom, Quantum Detection and Estimation Theory, Journal of Statistical Physics 1 (2) (1969) 231-252.
8. G. Ludwig, Attempt of an Axiomatic Foundation of Quantum Mechanics and More General Theories. III, Communications in Mathematical Physics 9 (1) (1968) 1-12.
9. E. B. Davies, J. T. Lewis, An Operational Approach to Quantum Probability, Communications in Mathematical Physics 17 (3) (1970) 239-260.
10. M. Ozawa, Quantum Reality and Measurement: A Quantum Logical Approach, Foundations of Physics 41 (3) (2011) 592-607.
11. M. Ozawa, Y. Kitajima, Reconstructing bohr's reply to epr in algebraic quantum theory, Foundations of Physics 42 (4) (2012) 475-487.
12. A. V. Zorin, V. V. Kuryshkin, L. A. Sevastianov, Description of the Spectrum of a Hydrogen-Like Atom, Bulletin of PFUR. Series Physics 6 (1998) 62-66.
13. A. V. Zorin, Approximate Determination of States in Quantum Mechanics of Kuryshkin, Bulletin of PFUR. Series Physics 12 (2004) 81-87, in Russian.
14. A. V. Zorin, Method of Investigation of Essential and Discrete Spectra of Hamilton-ian Operator of Hydogenlike Atom in Quantum Mechanics of Kuryshkin, Bulletin of PFUR. Series "Applied and Computer Mathematics" 3 (1) (2004) 121-131, in Russian.
15. A. V. Zorin, L. A. Sevastianov, N. P. Tretyakov, Computer Modeling of HydrogenLike Atoms in Quantum Mechanics with Nonnegative Distribution Function, Programming and Computer Software 33 (2) (2007) 94-104.
16. R. L. Stratonovich, On Distributions in Representation Space, Soviet Physics. JETP 4 (6) (1957) 891-898.
17. J. C. Varilly, J. Gracia-Bondia, The Moyal Representation for Spin, Annals of Physics 190 (1) (1989) 107-148.
18. M. Gadella, Moyal Formulation of Quantum Mechanics, Fortschritte der Physik 43 (3) (1995) 229.
19. I. I. Sobel'Man, R. A. Satten, An Introduction to the Theory of Atomic Spectra, Physics Today 27 (1974) 51.
20. L. A. Sevastyanov, A. V. Zorin, Lower Bound Estimating Method of Eigenvalues of Differentional Hamiltonian Operator in Quantum Mechanics of Kuryshkin, Bulletin of PFUR. Series "Applied and Computer Mathematics" 1 (1) (2002) 134-144, in Russian.
21. A. V. Zorin, L. A. Sevastianov, G. A. Belomestny, E. B. Laveev, Quantum Systems Modeling by Methods of Computer Algebra, in: Computer Algebra in Scientific Computing, St.Petersburg.-Munchen: TUM-Publ., 2004, pp. 497-506.
22. A. V. Zorin, L. A. Sevastianov, G. A. Belomestny, Numerical Search for the States with Minimal Dispersion in Quantum Mechanics with Non-Negative Quantum Distribution Function, in: Numerical Analysis and Its Applications, Springer, 2004, pp. 613-620.
23. A. A. Gusev, A. V. Zorin, K. P. Lovetsky, N. P. Tretyakov, Numerical Investigation of the Discrete Spectrum of Hamiltonian Operators of the Hydrogen-Like Atoms in Kuryshkin's Quantum Mechanics, Bulletin of PFUR. Series "Mathematics. Information Sciences. Physics" 3 (2007) 76-84, in Russian.
24. R. D. Richtmyer, Principles of Advanced Mathematical Physics, Vol. 1, Springer, 1979.
25. A. V. Zorin, Computer Implementation of Quantum Measurements Model, in: Information Technologies and Mathematical Modelling, ITMM 2015. 14th International Scientific Conference, named after A.F. Terpugov, Anzhero-Sudzhensk, Russia, 2015, pp. 273-275.
26. L. A. Sevastianov, A. V. Zorin, A. V. Gorbachev, A Quantum Measurements Model of Hydrogen-Like Atoms in Maple, in: Computer Algebra in Scientific Computing, Springer, 2013, pp. 369-380.
УДК 519.62; 530.145; 519.614
Модель квантовых измерений Курышкина-Вудкевича для атомов и ионов с одним валентным электроном
А. В. Горбачев, А. В. Зорин, Л. А. Севастьянов
Российский университет дружбы народов, Москва, Россия
Конструктивная форма модели квантовых измерений Курышкина-Вудкевича детально разработана для квантовой задачи Кеплера. Для более сложных квантовых объектов такая конструкция неизвестна. В то же время стандартная (не конструктивная) модель квантовых измерений Холево—Хелстрома годится для любого квантового объекта. Целью данной работы является распространение конструктивной модели квантовых измерений на более широкий класс квантовых объектов — модель квантовых измерений оптического спектра атомов и ионов с одним валентным электроном.
В работе модель Курышкина-Вудкевича, реализующая правило квантования Вейля— Курышкина в приложении к квантовой задаче Кеплера, распространена на квантовые системы с одним валентным электроном, например, на атомы щелочных металлов. В основу доказательства состоятельности модели положены две теоремы Като о компактных возмущениях операторов. В ходе доказательства получены явные формулы дискретного спектра валентного электрона для различных спектральных серий, зависящие от сериальных параметров возмущения спектра изолированного объекта в процессе квантовых измерений.
Ключевые слова: квантовые модели измерения, правила квантования, относительно компактное возмущение наблюдаемого оператора, возмущение наблюдаемого дискретного спектра
Литература
1. Zorin A. V., Sevastianov L. A. Hydrogen-Like Atom with Nonnegative Quantum Distribution Function // Physics of Atomic Nuclei. — 2007. — Vol. 70, issue 4. — Pp. 792-799.
2. Зорин А. В., Севастьянов Л. А. Модель квантовых измерений Курышкина-Вудкевича // Вестник РУДН. Серия «Математика. Информатика. Физика». — 2010. — Т. 3, вып. 1. — С. 98-103.
3. Sevastianov L. A., Zorin A. V., Gorbachev A. V. Pseudo-Differential Operators in the Operational Model of a Quantum Measurement of Observables // LNCS. — 2012. — Vol. 7125. — Pp. 174-181.
4. D'ariano G. M., Leonhardt U., Paul H. Homodyne Detection of the Density Matrix of the Radiation Field // Physical Review A. — 1995. — Vol. 52, No 3. — P. R1801.
5. D'ariano G. M. Measuring Quantum States // Quantum Optics and the Spectroscopy of Solids. — Springer, 1997. — Pp. 175-202.
6. Holevo A. S. Statistical Structure of Quantum Theory. — Springer Science & Business Media, 2003. — Vol. 67.
7. Helstrom C. W. Quantum Detection and Estimation Theory // Journal of Statistical Physics. — 1969. — Vol. 1, No 2. — Pp. 231-252.
8. Ludwig G. Attempt of an Axiomatic Foundation of Quantum Mechanics and More General Theories. III // Communications in Mathematical Physics. — 1968. — Vol. 9, No 1. — Pp. 1-12.
9. Davies E. B., Lewis J. T. An Operational Approach to Quantum Probability // Communications in Mathematical Physics. — 1970. — Vol. 17, No 3. — Pp. 239260.
10. Ozawa M. Quantum Reality and Measurement: A Quantum Logical Approach // Foundations of Physics. — 2011. — Vol. 41, No 3. — Pp. 592-607.
11. Ozawa M., Kitajima Y. Reconstructing Bohr's reply to EPR in algebraic quantum theory // Foundations of Physics. — 2012. — Vol. 42, No 4. — Pp. 475-487.
12. Zorin A. V., Kuryshkin V. V., Sevastianov L. A. Description of the Spectrum of a Hydrogen-Like Atom // Bulletin of PFUR. Series Physics. — 1998. — Vol. 6, issue 1. — Pp. 62-66.
13. Зорин А. В. Приближенное отыскание состояний в квантовой механике Курыш-кина // Вестник РУДН. Серия «Физика». — 2004. — Т. 12. — С. 81-87.
14. Зорин А. В. Метод исследования существенного и дискретного спектров оператора Гамильтона водородоподобного атома в квантовой механике Курышкина // Вестник РУДН. Серия «Прикладная и компьютерная математика». — 2004. — Т. 3, № 1. — С. 121-131.
15. Zorin A. V., Sevastianov L. A., Tretyakov N. P. Computer Modeling of HydrogenLike Atoms in Quantum Mechanics with Nonnegative Distribution Function // Programming and Computer Software. — 2007. — Vol. 33, No 2. — Pp. 94-104.
16. Stratonovich R. L. On Distributions in Representation Space // Soviet Physics. JETP. — 1957. — Vol. 4, No 6. — Pp. 891-898.
17. Varilly J. C., Gracia-Bondia J. The Moyal Representation for Spin // Annals of Physics. — 1989. — Vol. 190, No 1. — Pp. 107-148.
18. Gadella M. Moyal Formulation of Quantum Mechanics // Fortschritte der Physik. — 1995. — Vol. 43, No 3. — P. 229.
19. Sobel'Man I. I., Satten R. A. An Introduction to the Theory of Atomic Spectra // Physics Today. — 1974. — Vol. 27. — P. 51.
20. Зорин А. В., Севастьянов Л. А. Метод оценок снизу для собственных значений дифференциального оператора Гамильтона в квантовой механике Курышкина // Вестник РУДН. Серия «Прикладная и компьютерная математика». — 2002. — Т. 1, № 1. — С. 134-144.
21. Quantum Systems Modeling by Methods of Computer Algebra / A. V. Zorin, L. A. Sevastianov, G. A. Belomestny, E. B. Laveev // Computer Algebra in Scientific Computing / St.Petersburg.-Munchen: TUM-Publ. — 2004. — Pp. 497-506.
22. Zorin A. V., Sevastianov L. A., Belomestny G. A. Numerical Search for the States with Minimal Dispersion in Quantum Mechanics with Non-Negative Quantum Distribution Function // Numerical Analysis and Its Applications. — Springer, 2004. — Pp. 613-620.
23. Численное исследование дискретного спектра оператора Гамильтона водородоподобного атома в квантовой механике Курышкина / А. А. Гусев, А. В. Зорин, К. П. Ловецкий, Н. П. Третьяков // Вестник РУДН. Серия «Математика. Информатика. Физика». — 2007. — Т. 3, вып. 4. — С. 76-84.
24. Richtmyer R. D. Principles of Advanced Mathematical Physics. — Springer, 1979. — Vol. 1.
25. Zorin A. V. Computer Implementation of Quantum Measurements Model // Information Technologies and Mathematical Modelling, ITMM 2015. 14th International Scientific Conference, named after A.F. Terpugov / Anzhero-Sudzhensk, Russia. — 2015. — Pp. 273-275.
26. Sevastianov L. A., Zorin A. V., Gorbachev A. V. A Quantum Measurements Model of Hydrogen-Like Atoms in Maple // Computer Algebra in Scientific Computing / Springer. — 2013. — Pp. 369-380.
© Gorbachev A. V., Sevastianov L.A., Zorin A. V., 2016
