Expansion of the hydrogenoid atomic wave functions in 1. A besselian extension of the Taylor expansion, and its group structure

摘要

The series expansion of the hydrogenoid electronic wave functions in the inverse quantum number n~(-1) involves the besselian functions z~(n/2)J_n(2z~(1/2)). Such expansions are not restricted to the specific confluent hypergeometric functions which constitute the atomic radial wave functions, they can be applied to any regular function. The former functions are better understood as a particular case of the more general family of functions B_n(η,z) = (z/η)~(n/2)J_n[2(ηz)~(1/2)], with η any complex number. Like the monomials z~n! of the Taylor expansion, which are the particular case η = 0, the functions B_n are derivatives of one another (B'_(n-1) = B_n). The operators U_η which convert the monomials z~n! into the functions B_n(η,z) constitute a one complex parameter continuous group, corresponding to the group of multiplication operators e~(-η/p) in the space of Laplace transformed functions (Lf)(p).