danДоклады Национальной академии наук БеларусиDoklady of the National Academy of Sciences of Belarus1561-83232524-2431The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-8323-2019-63-3-282-290dan-613Research ArticleФИЗИКАPHYSICSСферические решения уравнения для частицы со спином 3/2Spherical solutions of the wave equation for a spin 3/2 particleИвашкевичА. В.IvashkevichA. V.

Ивашкевич Алина Валентиновна – студент

ул. Студенческая, 28, 247760, Мозырь, Гомельская область, Республика Беларусь

Ivashkevich Alina Valentinovna – Student

28, Studencheskaya Str., 247760, Mozyr, Gomel region, Republic of Belarus

ivashkevich.alina@yandex.byОвсиюкЕ. М.OvsiyukE. M.

Овсиюк Елена Михайловна – канд. физ.-мат. наук, доцент

ул. Студенческая, 28, 247760, Мозырь, Гомельская область, Республика Беларусь

Оvsiyuk Еlena Мikhailovna – Ph. D. (Physics and Mathematics), Assistant professor

28, Studencheskaya Str., 247760, Mozyr, Gomel region, Republic of Belarus

e.ovsiyuk@mail.ruКисельВ. В.KiselV. V.

Кисель Василий Васильевич – канд. физ.-мат. наук, доцент

ул. П. Бровки, 6, 220013, Минск, Республика Беларусь

Kisel Vasily Vasilyevich – Ph. D. (Physics and Mathematics), Assistant professor

6, P. Brovka Str., 220013, Minsk, Republic of Belarus

vasiliy_bspu@mail.ruРедьковВ. М.Red’kovV. M.

Редьков Виктор Михайлович – д-р физ.-мат. наук, гл. науч. сотрудник

пр-т Независимости, 68-2, 220072, Минск, Республика Беларусь

Red’kov Viktor Mikhailovich – Ph. D. (Physics and Mathematics), Chief researcher

68-2, Nezavisimosti Ave., 220072, Minsk, Republic of Belarus

v.redkov@dragon.bas-net.byМозырский государственный педагогический университет имени И. П. ШамякинаMozyr State Pedagogical University named after I. P. ShamyakinБелорусский государственный университет информатики и радиоэлектроникиBelarusian State University of Informatics and RadioelectronicsИнститут физики имени Б. И. Степанова Национальной академии наук БеларусиB. I. Stepanov Institute of Physics of the National Academy of Sciences of Belarus201927062019633282290Copyright © Ивашкевич А.В., Овсиюк Е.М., Кисель В.В., Редьков В.М., 20192019Ивашкевич А.В., Овсиюк Е.М., Кисель В.В., Редьков В.М.Ivashkevich A.V., Ovsiyuk E.M., Kisel V.V., Red’kov V.M.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://doklady.belnauka.by/jour/article/view/613

Волновое уравнение для частицы со спином 3/2, описываемой 16-компонентным вектор-биспинором, исследовано в сферической системе координат. В рамках подхода Паули–Фирца уравнение разбивается на основное и два дополнительных, алгебраическое и дифференциальное. Строятся решения, на которых диагонализируются четыре оператора: энергии, квадрата и третьей проекции полного момента, пространственного отражения, им соответствуют квантовые числа {ε, j, m, P}. После проведения разделения переменных выведена основная система из 8 зацепляющихся радиальных дифференциальных уравнений 1-го порядка и 4 условия связи: 2 алгебраических и 2 дифференциальных. Основная система приводится к виду 4 раздельных уравнений 2-го порядка, решения которых строятся в функциях Бесселя. С использованием свойств функций Бесселя вся система радиальных уравнений для частицы со спином 3/2 приведена к одному алгебраическому линейному уравнению A1a1 + A2a2 + A3a3 = 0 относительно величин a1, a2, a3, в котором коэффициенты Ai выражаются через квантовые числа ε, j. Выбраны наиболее симметричные решения, которые определяют два решения при фиксированных квантовых числах {ε, j, m, P}.

The wave equation for a spin 3/2 particle, described by 16-component vector-bispinor, is investigated in spherical coordinates. In the frame of the Pauli–Fierz approach, the complete equation is split into the main equation and two additional constraints, algebraic and differential. The solutions are constructed, on which 4 operators are diagonalized: energy, square and third projection of the total angular momentum, and spatial reflection, these correspond to quantum numbers {ε, j, m, P}. After separating the variables, we have derived the radial system of 8 first-order equations and 4 additional constraints. Solutions of the radial equations are constructed as linear combinations of the Bessel functions. With the use of the known properties of the Bessel functions, the system of differential equations is transformed to the form of purely algebraic equations with respect to three quantities a1, a2, a3. Its solutions may be chosen in various ways by solving the simple linear equation A1a1 + A2a2 + A3a3 = 0 where the coefficients Ai are expressed trough the quantum numbers ε, j. Two most simple and symmetric solutions have been chosen. Thus, at fixed quantum numbers {ε, j, m, P} there exists double-degeneration of the quantum states.

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The authors declare that there are no conflicts of interest present.