vestifmИзвестия Национальной академии наук Беларуси. Серия физико-математических наукProceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series1561-24302524-2415The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-2430-2026-62-1-48-58vestifm-875Research ArticleФИЗИКАPHYSICSНерелятивистское описание частицы со спином 3/2 в магнитном поле, тетрадные калибровочные преобразованияThe non-relativistic problem for a spin 3/2 particle in magnetic field, and tetrad gauge transformationsКузьмичА. М.KuzmichA. M.

Кузьмич Анастасия Михайловна – аспирант, младший научный сотрудник Центра фундаментальных взаимодействий и астрофизики

пр. Независимости, 68-2, 220072, Минск

Anastasia M. Kuzmich – Postgraduate Student, Junior Researcher of the Center of Fundamental Interactions and Astrophysics

68-2, Nezavisimosti Ave., 220072, Minsk

 

miss.nastya.01@list.ruИвашкевичА. В.IvashkevichA. V.

Ивашкевич Алина Валентиновна – научный сотрудник Центра фундаментальных взаимодействий и астрофизики

пр. Независимости, 68-2, 220072, Минск

Alina V. Ivashkevich – Researcher of the Center of Fundamental Interactions and Astrophysics

68-2, Nezavisimosti Ave., 220072, Minsk

ivashkevich.alina@yandex.byРедьковВ. М.Red’kovV. M.

Редьков Виктор Михайлович – доктор физико-математических наук, главный научный сотрудник Центра фундаментальных взаимодействий и астрофизики

пр. Независимости, 68-2, 220072, Минск

Viktor M. Red’kov – Dr. Sc. (Physics and Mathematics), Chief Researcher of the Center of Fundamental Interactions and Astrophysics

68-2, Nezavisimosti Ave., 220072, Minsk

v.redkov@ifanbel.bas-net.byИнститут физики имени Б. И. Степанова Национальной академии наук БеларусиB. I. Stepanov Institute of Physics of the National Academy of Sciences of Belarus2026310320266214858Copyright © Кузьмич А.М., Ивашкевич А.В., Редьков В.М., 20262026Кузьмич А.М., Ивашкевич А.В., Редьков В.М.Kuzmich A.M., Ivashkevich A.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://vestifm.belnauka.by/jour/article/view/875

Рассмотрена задача о нерелятивистской частице со спином 3/2 в магнитном поле с использованием цилиндрических координат и двух тетрад – декартовой и цилиндрической. Здесь возникают различные представления для 4-компонентной волновой функции: три декартовых  Lcart  Ψcart  ,Ψcart и получаемых с использованием калибровочного преобразования  Lcyl  = S (φ) L cart три цилиндрических  Lcyl   Ψcyl   ,Ψcyl  Сначала вводится нерелятивистское уравнение для частицы со спином 3/2 в базисах с недиагональной и диагональной третьей проекцией спина. Найдены решения двух типов, первый соотносится с диагонализацией оператора орбитального момента, а второй – с диагонализацией оператора третьей проекции полного углового момента. Возникающие при этом дифференциальные уравнения решены в терминах вырожденных гипергеометрических функций, найдены соответствующие спектры энергии. Вводится калибровочное преобразование, которое связывает декартову и цилиндрическую тетрады и позволяет преобразовать систему уравнений в полярной координате от декартовой тетрады к цилиндрической. Определены правила калибровочных преобразований для диагонализирующихся операторов полного и орбитального углового момента.

In the present paper, we will focus on the non-relativistic problem for a spin 3/2 particle in magnetic field, applying cylindrical coordinates and two tetrads: Cartesian and cylindrical. Here appear six different presentations for 4-component wave functions: three Cartesian ones Lcart  Ψcart  ,Ψcart  and provided by using the relevant gauge transformation Lcyl  = S (φ) L cart three different presentations in cylindrical tetrad Lcyl   Ψcyl   ,Ψcyl  First, we specify the non-relativistic equation for a spin 3/2 particle in magnetic field for Cartesian tetrad in bases with non-diagonal and diagonal matrix of the third spin projection. Solutions of two types are found: the first one is associated with the operator of the orbital angular momentum; the second solution relates to the third projection of the total angular momentum. Equations arising here are solved in terms of the confluent hypergeometric functions, and the corresponding energy spectra are found. The gauge transformation is introduced which relates two tetrads: Cartesian and cylindrical; it permits us to transform the system of equations in polar coordinate from Cartesian tetrad to cylindrical one. The rules for gauge transformations of diagonalized operators of the total angular and orbital momentums are found.

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