Structure of quantum mechanics
UDC
530.145.81DOI:
https://doi.org/10.31429/vestnik-17-2-66-73Abstract
Interest in quantum mechanics is growing in connection with new experiments with quantum particles and a deeper knowledge of nano- and subatomic principles. The practical use of quantum mechanic methods goes in parallel with the constant rethinking of its foundations, its ideological and epistemological role. A rigorous exposition of the mathematical foundations of quantum mechanics in abstract-Hilbert spaces was given in the book of J. Von Neumann in 1933. It is the first and only experiment that has been brought to the end, the presentation of the apparatus of quantum mechanics with the sequence and rigor which usually presented in a purely mathematical theory. Later in 1949 the group of French mathematicians under the pseudonym N. Bourbaki introduced the concept of "mathematical structure" into mathematics. To date, there is no narration of quantum mechanics, where mathematical rigor is combined with the concept of "mathematical structure". In the framework of the approach formulated by the group of N. Burbaki, quantum mechanics is a composite structure consisting of three simple mathematical structures: the Hilbert space of complex-valued vectors, the space of linear self-adjoint operators and the structure of classical mechanics are showed in this article.
Keywords:
quantum mechanics, axiomatic method, mathematical structure, Hilbert spaceReferences
- Serikov, A.A., Harkyanen, V.N. One-dimensional steady migration of quantum particles. Theoretical and mathematical physics, vol. 78, iss. 1. 1989, pp. 82–-88.
- Tan, W.-C., Inkson, J.C. Magnetization, persistent currents, and their relation in quantum rings and dots. Phys. Rev. B, vol. 60, 1999, pp. 5626–5635. DOI: 10.1103/PhysRevB.60.5626
- Margulis, V.A. Magnitnyy moment kol'tsa Volkano [The magnetic moment of the Volcano ring]. Fizika tverdogo tela [Solid State Physics], 2008, vol. 50 . pp. 148–153. (In Russian)
- Styer, D.F., Balkin, M.S., Becker, K.M., Burns, M.R., Dudley, C.E., Forth, S.T., Gaumer, J.S, Kramer, M.A., Oertel, D.C., Park, L.H., Rinkoski, M.T., Smith, C.T., Wotherspoon, T.D. Nine formulations of quantum mechanics. Am. J. of Phys., 2002, vol. 70, pp. 288–297.
- Belinskiy, A.V. Kvantovyye izmereniya [Quantum measurements]. BINOM, Moscow, 2008. (In Russian)
- Neumann, J. Mathematical of Quantum Mechanics, Berlin 1932.
- Bourbaki, N. L'Architecture des mathematiques. In: Les grands courants de la pensée mathématique, 1948, pp. 35–37.
- Kolmogorov, A.N. Matematika –- nauka i professiya [Mathematics as a science and a profession]. Nauka. Gl. red. fiz.-mat. lit., Moscow, 1988. (In Russian)
- Lebedev, K.A. O metodicheskikh i nauchnykh printsipakh sozdaniya shkol'nogo uchebnika matematiki serii ``MGU-shkolE''. I. Chisloviyye sistemy (5-6 klassy) [On the methodological and scientific principles of creating a school textbook of mathematics series ``MSU-school'' Pt. I. Numerical systems (grades 5-6)]. Matematicheskoye obrazovaniye [Mathematical education], 2016, no. 3(79), pp. 3–20. (In Russian)
- Lebedev, K.A. Arkhitektura elementarnoy matematiki [Architecture of elementary mathematics]. Kuban State University, Krasnodar, 2000. (In Russian)
- Varden, B.L. Algebra, vol. 2, Springer, New York, 1991.
- Lebedev, K.A. Arkhitektura matematiki. Topologiya, algebra i funktsional'nyy analiz [Architecture of elementary mathematics]. Kuban State University, Krasnodar, 2001. (In Russian)
- Makki, Dzh. Lektsii po matematicheskim osnovam Kvantovoy mekhaniki [Lectures on the mathematical foundations of quantum mechanics]. Mir, Moscow, 1965. (In Russian)
- Faddeyev, L.D., Yakubovskiy, O.A. Lektsii po kvantovoy mekhanike dlya studentov-matematikov [Lectures on quantum mechanics for mathematics students]. Izd-vo Leningr. un-ta, Leningrad, 1980. (In Russian)
- Landau, P.D., Lifshits, Ye.M. Kvantovaya mekhanika. Nerelyativistskaya teoriya [Quantum mechanics. Nonrelativistic theory]. Nauka, Moscow, 1989. (In Russian)
- Levich, V.G., Vdovin, Yu.A., Myamlin, V.A. Kurs teoreticheskoy fiziki. T. 2. Kvantovaya mekhanika, kvantovaya statistikami fizicheskaya kinetika [Course of theoretical physics. T.2. Quantum mechanics, quantum statistics, physical kinetics]. Nauka, Moscow, 1971. (In Russian)
- Shpol'skiy, E.V. Atomnaya fizika. Ch. 2 [Atomic physics. Part 2]. Mir, Moscow, 1974. (In Russian)
- Shiff, L. Kvantovaya mekhanika [Quantum mechanics]. Nauka, Moscow, 1959. (In Russian)
- Sadovnichiy, V.A. Teoriya operatorov [Theory of operators]. Vysshaya shkola, Moscow, 1999. (In Russian)
- Gel'fand, I.M. Lektsii po lineynoy algebre [Lectures on linear algebra]. Nauka, Moscow, 1971. (In Russian)
- Trenogin, V.A. Funktsional'nyy analiz [Functional analysis]. Fizmatlit, Moscow, 1980. (In Russian)
- Kolmogorov, A.N., Fomin, S.S. Elementy teorii funktsiy i funktsional'nogo analiza [Elements of function theory and functional analysis]. Nauka, Moscow, 1976. (In Russian)
- Kantorovich, L.V., Akilov, G.P. Funktsional'nyy analiz [Functional analysis]. Nauka, Moscow, 1977. (In Russian)
- Hilbert, D., Bernays, P. Grundlagen der Mathematik. II, Die Grundlehren der mathematischen Wissenschaften, 50, Berlin, New York: Springer-Verlag.
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