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"10.29235/1561-2430-2025-61-4-307-319vestifm-864Research ArticleФИЗИКАPHYSICSЗадача о собственных значениях обобщенного оператора спиральности для частицы со спином 3/2 в магнитном поле и метод проективных операторовEigenvalues of the generalized helicity operator for spin 3/2 particle in the presence of the magnetic field and the projective operators methodИвашкевичА. В.IvashkevichA. V.

Ивашкевич Алина Валентиновна – научный сотрудник

пр. Независимости, 68-2, 220072, Минск

Alina V. Ivashkevich – Researcher

68-2, Nezavisimosti Ave., 220072, Minsk 

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

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

пр. Независимости, 68-2, 220072, Минск

Viktor V. Red’kov – Dr. Sc. (Physics and Mathematics), Professor, Chief Researcher of the Center for 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 Belarus202506012026614307319Copyright © Ивашкевич А.В., Редьков В.М., 20262026Ивашкевич А.В., Редьков В.М.Ivashkevich A.V., Red’kov V.V.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/864

Решена задача о собственных значениях обобщенного оператора спиральности для частицы со спином 3/2 во внешнем однородном магнитном поле. После разделения переменных в уравнении на собственные значения в цилиндрической системе координат (r, ϕ, z) и соответствующей тетраде найдена система дифференциальных уравнений первого порядка в переменой r для 16 функций. Эта система решена на основе применения метода проективных операторов, построенных на основе третьей проекции оператора спина частицы. В соответствии с методом Федорова – Гронского все 16 переменных могут быть выражены только через 4 различающиеся функции, удовлетворяющие уравнениям вырожденного гипергеометрического типа. Дальнейшая задача сводится к анализу однородной алгебраической системы уравнений для 16 неизвестных величин. В итоге найдены уравнения 2-го и 4-го порядков, корни которых определяют собственные значения оператора спиральности.

The eigenvalue problem for generalized helicity operator for a spin 3/2 particle in presence of the uniform magnetic field is solved. After separating the variables in the basis of cylindrical coordinates (r, ϕ, z) and the tetrad, the system of 16 first-order differential equations in the variable r is derived. This system is studied with the use of the method of projective operators, constructed with the use of the third projection of the spin for the particle. In accordance with thе method by Fedorov – Gronskiy, all 16 variables may be expressed in terms of only 4 distinguished functions, which are constructed in terms of confluent hypergeometric functions. Further the problem reduces to studying the linear algebraic homogeneous system for 16 algebraic variables. In the end, we derive algebraic equations of the second and the fourth order, their roots determine the possible eigenvalues of the helicity operator.

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