Wigner D-functions

Eigenfunctions of the operator M are obviously proportional to Wigner D-functions. Therefore a general solution (2.81) with initial condition (2.80) has to be represented as... 

If one performs a turn of the coordinate system x y z by Euler angles a, 0,7, thus transferring it to the position xyz, then the components Ay and A of the vector A in the new coordinate system may be expressed through the components Ay>, A11 in the initial coordinate system by means of Wigner D-functions... 

The spherical functions may be expressed through the Wigner D-function ... 

The authors are hopeful that the book will serve as a practical manual for the researcher, both theorist as well as experimentalist, in atomic, molecular or chemical physics. With this in mind, we have paid special attention to the comprehensive information necessary to follow all calculations through to their conclusions. We also hope that the material in the appendices will take the place of a short handbook with formulae on vector calculus, spherical functions, Wigner D-functions, Clebsch-Gordan coefficient tables, etc. We fully realize that one may find in various pa-... 

The total wave function tyJM with angular momentum J and its projection M can be expanded in terms of Wigner D-functions [17, 18] ... 

Wigner D-functions are defined using the convention of Rose (11) thus. 

Because the elements of the amplitude matrix can be expressed in terms of the elements of the transition matrix, the above relation can be used to express the elements of the amplitude matrix as functions of the particle orientation angles Op, /3p and 7p. The properties of the Wigner D-functions can then be used to compute the integrals over the particle orientation angles. 

The Wigner d-functions are real and have the following symmetry properties [58] ... 

The orthogonality property of the Wigner d-functions is similar to that of the associated Legendre functions and is given by... 

The Wigner d-functions are related to the generalized spherical functions Pmm relation [78]... 

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