Approximate solution of the strongly magnetized hydrogenic problem with the use of an asymptotic property

It is shown that the effective potentials of the adiabatic approximation, which depend on the magnetic field parameter $\beta =\frac{B}{B_0}$ ($B_0\cong 4.70\times {10}^5$ T) and the quantum number $s=-m\ge 0$ of the $z$ component of the angular momentum, asymptotically (i.e., for large $s$) can be traced back to one single potential function which solely depends on the ratio $p=\frac{\beta }{(s+\frac{1}{2})}$. For this asymptotic potential, numerical solutions of the Schrödinger equation are determined in the range ${10}^{-3\mathrm{}}\le p\le {10}^{+3\mathrm{}}$ for $0\le \nu \le 20$ ($\nu$ being the number of nodes of the longitudinal wave function). Exploiting the concept of quantum excesses, the asymptotic energies are extrapolated to $\nu >20\mathrm{}$. It is found that the asymptotic energies provide the energy values of the real physical problem of hydrogenic atoms in magnetic fields $\beta \gtrsim 1$ within an accuracy of ≲1% for every $s\ge 3$ and arbitrary $\nu$, with the accuracy improving rapidly, as $\beta$, $s$, or $\nu$ is increased. Thus our results ideally complement those for $s<\mathrm{}3\mathrm{}$ for which accurate results have been tabulated in the literature.