Asymptotic methods for Rydberg transitions

Quantum mechanical expressions of several important physical quantities like the hydrogen dipole matrix elements, line strength and form factors for several excitation processes have long been available in the literature in terms of the terminating hypergeometric functions but calculations from these expressions generally present serious numerical problems for large principal quantum numbers nn and n′n′. Determination of asymptotic and other appropriate approximations of these quantities for large nn and n′n′ has for long been posing challenge. We discuss a recent method that transforms the terminating hypergeometric functions into the Jacobi polynomials and exploits the properties of the Jacobi polynomials to provide a solution to the problem of the evaluation of these physical quantities. A noteworthy result of this method is that the exploitation of the recurrence relation of the Jacobi polynomials permits exact numerical calculations for various Rydberg processes for so large n,n′∼2000n,n′∼2000 for which computation was usually not possible earlier. Another noteworthy outcome is that the method readily leads to a strikingly accurate expression of a Rydberg matrix element between nearby Rydberg states in terms of the Bessel functions, called an NRS-formula, which also helps to solve, without any recourse to classical and semiclassical arguments, a long standing problem of how to consistently derive the formula of classical mechanics obtained earlier by invoking the correspondence principle. The numerical results from the exact and approximate formulae of various quantum matrix elements presented in the article are so extensive that they reveal how various formulae including those of the correspondence principle convergence to their respective exact quantum expressions. Also, for numerical and analytical study of various matrix elements involving states lying near the continuum threshold, which have posed problems earlier, simpler quantum expressions are presented. The quantum expressions presented in the article provide nearly complete solutions over the hydrogenic bound-state spectrum for the calculation of several physical quantities like the radial dipole matrix element, line strength and form factors for transitions between arbitrary ss states and between arbitrary circular states.