Normal Ordering of the Angular Momentum Cylindrical Ladder Operators and their Products with Wigner D-functions

Xuanhao, C.; Millionshchikov, D.V.; Efremov, I.M.; Krasnoshchekov, S.V.

Авторы: Xuanhao Chang, Millionshchikov Dmitry V., Efremov Ilya M., Krasnoshchekov Sergey V.
Журнал: Journal of Chemical Physics
Год издания: 2023
Издательство: AIP Publishing
Местоположение издательства: United States
Первая страница: 1
DOI: 10.1063/5.0142809
Аннотация: The operator canonical perturbation theory (CPT) is an efficient tool for solving the molecular vibration-rotation Schrödinger equation. The corresponding Watson Hamiltonian can be written using angular momentum cylindrical ladder operators (Jz,J±=Jx∓iJy) possessing the Lie algebra su(2) commutation relations [J+,J−]=2Jz, [Jz,J±]=±J±. The reduced effective Hamiltonians suitable for fitting to observed spectra are traditionally based on Hermitian basis sets, e.g. (J2)l[Jzn, J+m + J−m)]+. It is beneficial to re-express such Hamiltonians using sums of normal ordered products of ladder operators Jza J+b J−c. For instance, in the CPT the unitary transformations reduce to the normal ordering problem. Similarly, the dipole moment transition probabilities can be evaluated using Wigner functions D10,ε(ε=0,±1) possessing complex commutation relations with Jα-operators. The related line strengths are proportional to the squared matrix elements of unitary transformed dipole moment operator given by a polynomial in normal ordered products D1 0,ε Jza J+b J−c. We have applied the technique of combinatorial calculations associated with the classical representation theory and obtained compact formulas for normal ordering of the typical raw products Jza J+b J−c Jzd J+e J−f and Jza J+b J−c D1 0,ε. The theory of universal enveloping Lie algebras and representation theory show that the resulting formulas cannot be further improved from the mathematical point of view. The results have a wide scope of applications in other fields of physics including other operators with the same commutation properties. The resulting expressions for normal orderings and routines for evaluation of matrix elements, as well as systematic tests of working formulas were coded in Fortran and the software is available through the GitHub hosting service.
Добавил в систему: Краснощеков Сергей Вадимович

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