Abstract
In this paper, the orbits of observable physical quantities of position and momentum operators at the states of quantum Hilbert spaces are created. Also the Hilbert space of finite orbits and the Fréchet–Hilbert space of all orbits are created and the orbital operators corresponding to these observable operators in these spaces of orbits are defined and studied. We call this process the orbitization, and the obtained model the orbital quantum mechanics. The orbitization process is compared with the quantization process. The generalization of well-known canonical commutation relations for orbital operators corresponding to the position and momentum operators is established. Also, the Heisenberg uncertainty principle for the orbital operators is proved and the question of achieving equality in the Heisenberg inequality is considered.
References
[1] J. Becnel and A. Sengupta, The Schwartz space: Tools for quantum mechanics and infinite dimensional analysis, Mathematics 3 (2015), no. 2, 527–562. 10.3390/math3020527Search in Google Scholar
[2] S. Dierolf and K. Floret, Über die Fortsetzbarkeit stetiger Normen, Arch. Math. (Basel) 35 (1980), no. 1–2, 149–154. 10.1007/BF01235333Search in Google Scholar
[3] S. Dierolf and D. N. Zarnadze, On homomorphisms between locally convex spaces, Note Mat. 12 (1992), 27–41. Search in Google Scholar
[4] B. C. Hall, Quantum Theory for Mathematicians, Grad. Texts in Math. 267, Springer, New York, 2013. 10.1007/978-1-4614-7116-5Search in Google Scholar
[5] E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley &Sons, New York, 1978. Search in Google Scholar
[6] E. Prugovečki, Quantum Mechanics in Hilbert Space, 2nd ed., Pure Appl. Math. 92, Academic Press, New York, 1981. Search in Google Scholar
[7] M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, Academic Press, New York, 1972. Search in Google Scholar
[8] S. Rolewicz, On orbits of elements, Studia Math. 32 (1969), 17–22. 10.4064/sm-32-1-17-22Search in Google Scholar
[9] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Math. Libr. 18, North-Holland, Amsterdam, 1978. Search in Google Scholar
[10] S. Tsotniashvili and D. Zarnadze, Selfadjoint operators and generalized central algorithms in Frechet spaces, Georgian Math. J. 13 (2006), no. 2, 363–382. 10.1515/GMJ.2006.363Search in Google Scholar
[11] D. Ugulava and D. Zarnadze, Generalized spline algorithms and conditions of their linearity and centrality, Proc. A. Razmadze Math. Inst. 160 (2012), 143–164. Search in Google Scholar
[12] D. Ugulava and D. Zarnadze, On a linear generalized central spline algorithm of computerized tomography, Proc. A. Razmadze Math. Inst. 168 (2015), 129–148. Search in Google Scholar
[13] D. Ugulava and D. Zarnadze, Approximate solution of Schrödinger equation in the spaces of orbits, XI Annual Meeting of the Georgian Mechanical Union, Batumi Shota Rustaveli State University, Batumi (2020), http://gmu.gtu.ge/Batumi2020. Search in Google Scholar
[14] D. Ugulava and D. Zarnadze, About the concept of orbital quantum mechanics, XI Annual Meeting of the Georgian Mechanical Union, Batumi Shota Rustaveli State University, Batumi (2021), http://gmu.gtu.ge/Batumi2021/Conference_Batumi_2021+.pdf. Search in Google Scholar
[15] D. Ugulava and D. Zarnadze, On a central algorithm for calculation of the inverse of the harmonic oscillator in the spaces of orbits, J. Complexity 68 (2022), Paper No. 101599. 10.1016/j.jco.2021.101599Search in Google Scholar
[16] D. Ugulava and D. Zarnadze, On canonical commutation relation for orbital operators corresponding to Creation and Annihilation operators, XII International Conference of the Georgian Mathematical Union, Batumi Shota Rustaveli State University, Batumi (2022), http://gmu.gtu.ge/Batumi2022/Conference_Batumi_2022.pdf. Search in Google Scholar
[17] D. Ugulava and D. Zarnadze, On linear spline algorithms of computerized tomography in the space of n-orbits, Georgian Math. J. 29 (2022), no. 6, 939–952. 10.1515/gmj-2022-2185Search in Google Scholar
[18] D. Ugulava and D. Zarnadze, Orbitization of quantum mechanics, GESJ Comput. Sci. Telecommun. 2022 (2022), no. 1(61), 59–63. Search in Google Scholar
[19]
S.-A. Wegner,
Universal extrapolation spaces for
[20] D. N. Zarnadze, Fréchet spaces with some classes of proximinal subspaces (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 4, 711–725; translation in Math. USSR, Izv. 29 (1987), 67–79. Search in Google Scholar
[21] D. N. Zarnadze, Some topological and geometric properties of Fréchet–Hilbert spaces (in Russian), Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 5, 1001–1020; translation in Russian Acad. Sci. Izv. Math. 41 (1993), no. 2, 273–288. Search in Google Scholar
[22] D. N. Zarnadze, On a generalization of the least squares method for operator equations in some Fréchet spaces (in Russian), Izv. Ross. Akad. Nauk Ser. Mat. 59 (1995), no. 5, 59–72; translation in Izv. Math. 59 (1995), no. 5, 935–948. Search in Google Scholar
[23] D. Zarnadze and S. A. Tsotniashvili, Generalization of the canonical commutative relation in the quantum Fréchet–Hilbert space, XI International Conference of the Georgian Mathematical Union, Batumi Shota Rustaveli State University, Batumi (2021), http://gmu.gtu.ge/Batumi2021/Conference_Batumi_2021+.pdf. Search in Google Scholar
[24] D. Zarnadze, D. Ugulava, M. Kublashvili and P. Tsereteli, On calculation of the inverse of multidimensional harmonic oscillator on Schwartz space, South Caucasus Computing and Technology Workshop, Georgian Technical University, Tbilisi (2016), https://indico.cern.ch/event/572800/contributions/2319215/attachments/1350183/2037830/new_report.atlas610_1.pdf. Search in Google Scholar
[25] D. Zarnadze, D. Ugulava and S. Tsotniashvili, On canonical commutation relation for Creation and Annihilation operators in the strict Fréchet–Hilbert space of states, XII International Conference of the Georgian Mathematical Union, Batumi Shota Rustaveli State University, Batumi (2022), http://gmu.gtu.ge/Batumi2022/Conference_Batumi_2022.pdf. Search in Google Scholar
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Articles in the same Issue
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- On left and right Browder elements in Banach algebra relative to a bounded homomorphism
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- On normal partial isometries
- Gauss--Newton--Kurchatov method for the solution of non-linear least-square problems using ω-condition
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- Weighted generalized Moore–Penrose inverse
- On two extensions of the annihilating-ideal graph of commutative rings
- Umbral treatment and lacunary generating function for Hermite polynomials
- A generalization of the canonical commutation relation and Heisenberg uncertainty principle for the orbital operators
Articles in the same Issue
- Frontmatter
- On left and right Browder elements in Banach algebra relative to a bounded homomorphism
- Uniform excess of g-frames
- On normal partial isometries
- Gauss--Newton--Kurchatov method for the solution of non-linear least-square problems using ω-condition
- Maps preserving the bi-skew Jordan product on factor von Neumann algebras
- Boundary contact problems with regard to friction of couple-stress viscoelasticity for inhomogeneous anisotropic bodies (quasi-static cases)
- On uniform statistical convergence
- Approximate solution of the optimal control problem for non-linear differential inclusion on the semi-axes
- Energy-based localization of positive solutions for stationary Kirchhoff-type equations and systems
- e-reversibility of rings via quasinilpotents
- Weighted generalized Moore–Penrose inverse
- On two extensions of the annihilating-ideal graph of commutative rings
- Umbral treatment and lacunary generating function for Hermite polynomials
- A generalization of the canonical commutation relation and Heisenberg uncertainty principle for the orbital operators