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)8660ArticlesСтатьиProperties of Wigner Distribution Functions Applied to Quantum MechanicsСвойства квантовой функции распределения Вигнера в применении к квантовой механикеGorbachevA VГорбачёвАлександр Владимирович ; Peoples Friendship University of RussiaКафедра теоретической физики; Российский университет дружбы народовalexarus1986@gmail.comPeoples Friendship University of RussiaРоссийский университет дружбы народов150220122NO2 (2012)№2 (2012)788608092016Copyright ©; 2012, Горбачёв А.В.2012Горбачёв А.В.http://creativecommons.org/licenses/by/4.0https://journals.rudn.ru/miph/article/view/8660An operational model of quantum measurements was presented befor. In order to obtain constructive theoretical results from this model there is a need to define previously not described properties of Wigner distribution functions. The report contains the proof of these properties. Multidimensional generalization and relationships with different conventions of the Fourier transform were described.В процессе построения операциональной модели квантовых измерений возникла необходимость установить ряд ранее не описанных свойств квантовой функции распределения Вигнера. Данная работа посвящена доказательству этих свойств, так как они необходимы для получения ряда конструктивных теоретических результатов. Сделано обобщение на многомерный случай и показана зависимость от выбора формы записи преобразования Фурье.Wigner distribution functionsoperational model of quantum measurementquantum distribution functionквантовая функция распределения Вигнераоперациональная модель квантовых измеренийквантовая функция распределения1.Braginsky V.B., Vorontsov Y.I., Halily F.Y. Quantum Features of the Ponderomotive Meter of Electromagnetic Energy // Journ. Exper. Theor. Phys. - 1977. - Vol. 73. - Pp. 1340-1343.2.Braginsky V.B., Vorontsov Y.I., Halily F.Y. Optimal quantum measurements in detectors of gravitational radiation // Lett. Journ. Exper. Theor. Phys. - 1978. - Vol. 27. - Pp. 296-301.3.Holevo A.S. Statistical Structure of Quantum // Lecture Notes in Phys. - Berlin: Springer, 2001. - Vol. 67.4.Helstrom C.W. Quantum Detection and Estimation Theory. - New York etc.: Academic Press, 1976.5.Sevastianov L.A., Zorin A.V. The Method of Lower Bounds for the Eigenvalues of the Hamiltonian Differential Operator in Quantum Mechanics of Kuryshkin // Bull. PFUR, ser. "Appl. And Comp. Math.". - 2002. - Vol. 1, issue 1. - Pp. 134- 144.6.Zorin A.V., Sevastianov L.A. Hydrogen-Like Atom with Nonnegative Quantum Distribution Function // Physics of Atomic Nuclei. - 2007. - Vol. 70, issue 4. - Pp. 792-799.7.Sevastianov L.A., Zorin A.V., Gorbachev A.V. Pseudo-Differential Operators in the Operational Model of a Quantum Measurement of Observables // Lecture Notes in Computer Science. - 2012. - Vol. 7125. - Pp. 174-181.8.Kuryshkin V.V. On the Construction of Quantum Operators // Izv. VUZov. Phys. - 1971. - Issue 11. - Pp. 102-106.9.Wodkiewicz K. Operational Approach to Phase-Space Measurements in Quantum Mechanics // Phys. Rev. Lett. - 1984. - Vol. 52.10.Wigner E.P. On the Quantum Correlation for Thermodynamic Equilibrium // Phys. Rev. - 1932. - Vol. 40. - Pp. 749-759.11.Weyl H. Quantenmechanik und Gruppentheorie // Z. Phys. - 1927. - Vol. 46. - Pp. 1-39.12.Gorbachev A.V. - Modeling of Alkaline Metal Spectra using Quantum Mechanics with Nonnegative Quantum Distribution Function (in russian). - Master's thesis, PFUR, 2010.13.Richtmyer R.D. Principles of Advanced Mathematical Physics. - New York etc.: Springer-Verlag, 1978.