On the spectrum, radial wave functions, and hyperfine splittings of the Rydberg states in heavy alkali atoms

TL;DR

This paper numerically calculates the Rydberg states of heavy alkali atoms using spectral collocation, compares results with quasiclassical and WKB methods, and derives analytical approximations for wave functions and hyperfine splittings.

Contribution

It introduces a high-accuracy spectral collocation method for Rydberg states and develops new analytical approximations for wave functions and hyperfine constants.

Findings

Numerical spectrum matches well with quasiclassical results for low l states.

Discrepancies occur for l=3 states due to potential anomalies.

Analytical wave function approximations validate hyperfine constant calculations.

Abstract

We calculate the bound state spectrum of the highly excited valence electron in the heavy alkali atoms solving the radial Schr\"odinger eigenvalue problem numerically with an accurate spectral collocation algorithm that applies also for a large principal quantum number $n\gg1$. As an effective single-particle potential we favor the reputable potential of Marinescu \emph{et al}., {[}Phys. Rev.A \textbf{49}, 982(1994){]}. Recent quasiclassical calculations of the quantum defect of the valence electron agree for orbital angular momentum $l=0,1,2,...$ overall remarkably well with the results of the numerical calculations, but for the Rydberg states of rubidium and also cesium with $l=3$ this agreement is less fair. The reason for this anomaly is that the potential acquires for $l=3$ deep inside the ionic core a tiny second classical region, thus invalidating a standard WKB calculation with two widely spaced turning points. Comparing then our numerical solutions of the radial Schr\"odinger eigenvalue problem with the uniform analytic WKB approximation of Langer we observe everywhere a remarkable agreement, apart from a tiny region around the inner turning point. With help of an ansatz proposed by Fock we obtain for the \emph{s}-states a second uniform analytic approximation to the radial wave function complementary to the WKB approximation of Langer, which is exact for $r\to0^{+}$. The value of the radial \emph{s}-wave function at $r=0$ is analytically found, thus validating the Fermi-Segr\`e formula for the magnetic dipole interaction constant $A_{n,j,0}^{\left(\mathrm{HFS}\right)}$.