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3 the Generalised Schrödinger Equation Editorial Board Editor in Chief Mark Zilberman, MSc, Shiny World Corporation, Toronto, Canada Scientific Editorial Board Viktor Andrushhenko, PhD, Professor, Academician of the Academy of Pedagogical Sciences of Ukraine, President of the Association of Rectors of pedagogical universities in Europe John Hodge, MSc, retired, USA Petr Makuhin, PhD, Associate Professor, Philosophy and Social Communications faculty of Omsk State Technical University, Russia Miroslav Pardy, PhD, Associate Professor, Department of Physical Electronics, Masaryk University, Brno, Czech Republic Lyudmila Pet'ko, Executive Editor, PhD, Associate Professor, National Pedagogical Dragomanov University, Kiev, Ukraine Volume 8, Number 2 Publisher : Shiny World Corp. Address : 9200 Dufferin Street P.O. Box 20097 Concord, Ontario L4K 0C0 Canada E-mail : [email protected] Web Site : www.IntellectualArchive.com Series : Journal Frequency : Every 3 months Month : April - June 2019 ISSN : 1929-4700 DOI : 10.32370/IA_2019_02 Trademark © 2019 Shiny World Corp. All Rights Reserved. No reproduction allowed without permission. Copyright and moral rights of all articles belong to the individual authors. Physics M. Pardy The Complex Mass from Henstock-Kurzweil-Feynman-Pardy Integral ……………. 1 John C. Hodge STOE beginnings ……………………………………………………………….…………. 11 John C. Hodge STOE Explains "Planet 9" ………………………………………………………………… 13 John C. Hodge STOE Explanation for the ``Ether wind" ………………………………………………… 15 V. Litvinenko , A. Shapovalov, Low Inductive and Resistance Energy Capacitor …….………….………………….… 19 A. Ovsyannik Astronomy W. Duckss Why do Hydrogen and Helium Migrate from Some Planets and Smaller Objects? ... 24 Technics Modern Reading of the 40 Principles of TIPS Using the Example of the Technology V. Korobov for Activation of Fuel Mixtures ……………………………………………………………. 32 Computer Science A. Kravchenko The Practical Side of IoT Implementation in Smart Cities …………………………..... 52 Economics Workforce Training and Management Challenges in the Contemporary Smart A. Kravchenko Manufacturing (SM) …………………………………………………….………………..... 59 A. Kravchenko How Large Businesses Convert Volatility in Emerging Markets into Opportunity ..... 66 Law A. Kofanov, V. Anufriieva- Injukova, Theoretical and Practical Aspects of Arms Trafficking ………………………………… 74 M. Kulyk, Y. Tereshchenko, R. Kutsyi A. Kofanov, The Global Problem of the Third Millennium: Organized Transnational Cybercrime P. Bilenchuk, (Historiography) ……………………………………………………………………………. 86 O. Kofanova continued (continued) N. Kofanova, A. Kofanov, Decentralization of Authorities and Local Government Reform in Ukraine: Their E. Svoboda, Impact on the Efficiency of Public Law, Power and Strengthening of Democracy …. 91 N. Pavlovska, O. Romanenko Philosophy Olajiga Isaac Making Sense of the Cultural Other in the Fabric of Knowledge Vis-à-vis The Oladeji Yoruba Cultural Framework ……………………………………………………………… 108 L. Stoyanova Conceptual Ideas About the Creative Elite as a Creator of Creative Social Space ... 123 Socio – Phylosophical Creation of the Concept of “Soul”: the Experience of V. Cuba Postmodern Reflection ……………………………………………………………………. 132 The Foundations of Social and Personal in Archetypes N. Zaytseva in the Concept by K. G. Jung: A Phenomenological and Semiotic Presentation …… 144 Arts Education Social – Cultural Deviation of the Creative Environment: a Conceptual Approach to V. Rudoy the Phenomenon ………………………………………………………………..……….. 152 P. Naguliak, Traditional and Modern Methods of Painting on the Plener of Students of Art V. Rudoy Education …………………………………………………………………………………… 159 Education The Forming English Lexical Competence in Dialogic Speaking for Prospective I. Skril Experts of Hospitality and Restaurant Service …………………………………………. 170 L. Vorobiova Implementation of the Dalton Plan Ideas into Ukrainian Alternative Schooling …….. 175 R. Vdovychenko, Competence Approach to Student Youth Education …………………………………. 179 K. Shapochka Intensification of Cadet’s Foreign Language Competence Formation Through the L. Hrebeniuk Role-Playing Games ………………………………………………………………………. 183 O. Matviienko, Criteria, Indicators and Levels Future Primary School Teacher’s Readiness to T. Tutova Collaboration with Heterogeneous Groups of Students ………………………………. 190 Features of the Development of the Effective and Reflective Component of O. Voichun Students’ Civic Engagement at the University …………………………………………. 201 The Basic Methodological Approaches to Vocal-Choral Teaching Children of A. Hohol Primary School Age ……………………………………………………………………….. 211 The Practical Aspects of Conscious Attitude Formation to Fatherhood in Conditions K. Fediushkina of Vocational Educational Institutions …………………………………………………… 220 Social Work The System-Oriented Perspectives of Inculcating Innovative Technologies for I. Savelchuk Social Work ………………………………………………………………………………… 228 Book presentation O. Perova Olga Perova. “Art of Prywinko” …………………………………………………………... 235 Manuscript Guidelines …………………………………………………………………….. 236 DOI 10.32370/IA_2019_02_1 The complex mass from Henstock-Kurzweil-Feynman-Pardy integral Miroslav Pardy Department of Physical Electronics Masaryk University Kotl´aˇrsk´a2, 611 37 Brno, Czech Republic e-mail:[email protected] Abstract The Feynman integral is generalised so as to involve the random fluctuations of vacuum, from this integral the generalized Schr¨odingerequation is derived and the energy spectrum for the Coulomb potential determined. 1 Introduction The purpose of this paper is to generalize the Feynman integral over paths in the case, where we consider random fluctuations of vacuum, and to derive the generalized Schr¨odingerequation. The article is the improved version of the author article (Pardy, 1973), where the Henstock-Kurzweil integral (Henstock, 1946; Morales, et al., 2017) is not involved in it. 1 IntellectualArchive Vol. 8, No. 2, April - June 2019 1 First, let us remember Feynman's fictitious experiment by means of which he arrives at the integral over paths (Feynman & Hibbs, 1965). A source of electrons is considered in this experiment and a movable detector is fixed at a certain distance from the source. Between the source and the detector is put a screen with two holes, 1 and 2, in it. Now, we investigate the probability that an electron will arrive in the detector at various vertical distances x from the source. If we block hole 2, the particle will pass through hole 1, and if we block hole 1 the particle will go through hole 2. The probability P1(x) to find a particle at point x will be given in the first case like this: 2 P1(x) = jϕ1(x)j ; (1) where ϕ1(x) is the complex function called the probability amplitude corresponding to the path going through hole 1 in the detector. In the second case the probability to find a particle at point x is P2(x) and it holds good: 2 P2(x) = jϕ2(x)j ; (2) where ϕ2(x) is the probability amplitude corresponding to the path going through hole 2 in the detector. If we leave open both holes, the probability of finding a particle at point x will be given like this: 2 P (x) = jϕ1(x) + ϕ2(x)j : (3) Now, let us consider that there is not only one screen between the source and the detector but that there are N screens, each of them having M holes, with width ∆(Ai;Bi)j, where Ai, Bi are x-coordinates of the borders of the holes, j being an index belonging to the jth screen, j = 1, 2, ··· , N and i being an index belonging to the ith hole, i = 1, 2, ···, M; ∆ denotes the length of the interval ( , ). We can say that the trajectory of the particle rises in such a way that the particle goes through interval (a1; b1) at the time t1 , through the interval (a2; b2) at the time t2 ··· , through the interval (an; bn) at the time tn, where (ai; bi) are intervals from the set of intervals (Ai;Bi)j. According to Feynman, the probability amplitude ϕ(x) belongs to this trajectory. If we choose another set of numbers tk,(ak; bk), we get other alternative path of the particle. It is obvious that there are a great many of these alternative trajectories, and therefore the total amplitude for the process is the sum of the amplitudes for each route considered separately. For the case N ! 1, 2 IntellectualArchive Vol. 8, No. 2, April - June 2019 2 M ! 1, all trajectories will be continuous functions of the time t, and the total probability amplitude for the transition of a particle from point a to point b is as follows: X U(b; a) = ϕ[x(t)]; (4) a!b P where a ! b in means summation over all trajectories from a to b and x(t) is the x-coordinate of a path. The amplitude is postulated as follows: 1 ϕ[x(t)] = expf(i=h¯)S[x(t)]g; (5) A where Z t S = b L[_x(t); x(t)]dt (6) ta is the classical Hamilton-Jacobi action function and L is the Lagrange function. Feynman & Hibbs (1965) have shown that the probability amplitude U(x; x0) concerning the transition of the particle from point x0 to point x satisfies the integral equation Z U(x; x0) = U(x; x1)U(x1; x0)dx1: (7) If we put U(x; x0) = (x);U(x1; x0) = (x1), we get Z (x) = U(x; x1) (x1)dx1; (8a) or, generally Z (x; t) = U(x; t; x1; t1) (x1; t1)dx1 (8b) and the function (x; t) may be interpreted as the wave function of quantum mechanics. For an infinitesimal transition from point xk and time kk in point xk+1 and time tk+1, we can write (Blokhintsev, 1966): 1 U(x ; k ; x ; t ) = expf(i=h¯)S[x ; x ; ∆t]g; (9) k+1 k+1 k k A k+1 k where " ! # m x − x S[x ; x ; ∆t] = k+1 k − V (x )∆t ∆t; (10) k+1 k 2 ∆t k+1 where ∆t = tk+1 − tk; tk+1 > tk. 3 IntellectualArchive Vol. 8, No. 2, April - June 2019 3 2 Random fluctuations of the Vacuum in the integral over paths Let us notice term eq. (4) X U(x1; x0) = ϕ[x(t)]: (11) x0!x We can see that all amplitudes exp (i=h¯)S[x(t)] are multiplied only by a constant 1=A. In other words, the Feynman sum is constructed in such a way that all trajectories are considered equivalent and therefore equally probable. The question arises, as to how the formulation of the problem must be changed if we include the random fluctuations of the vacuum.
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