Radial power-like potentials: from the Bohr-Sommerfeld $\ell \neq 0$ state energies to the exact ones

Abstract

Two new Bohr-Sommerfeld (B-S) type quantization conditions for $d$-dimensional radial potentials are proposed for non-singular and singular potentials, respectively. The power-like potentials \hbox{$V=\operatorname{sign}(m)\,r^m$} for $m=-3/2,-1,-1/2, 1,3,4,6$ and $V=\log r$ (that corresponds to $m=0$) are studied in detail for radial quantum numbers $n_r \in [0,100]$ and for angular momenta $\ell=1,2,3,4,5$ in dimensions $d=2,3$. It is shown that for large $n_r$ the B-S energies $E_{BS}$ always tend to the exact ones. For $m=2$ (the radial harmonic oscillator) and $m=-1$ (the $d$-dimensional analogue of the $3D$ Coulomb potential) the B-S spectrum coincides with the exact one for the first and second quantization conditions, respectively. In general, up to a small parametric shift of the radial quantum number, the analytic formulas for the energy spectrum can be written for arbitrary power-like potential. A comparison of $E_{BS}$ with the energies $E^{(L)}_{BS}$, obtained for the radial potentials modified by the Langer correction for $d=3$, are in good agreement in general: in detail, it was checked for the quartic oscillator $V=r^4$. It is demonstrated that introducing a dimensionless {\it WKB correction} $γ$ into the r.h.s. of the proposed B-S type quantization condition allows us to reproduce the exact energies with $γ$ being very small for positive degrees $m \geq 0$ and sufficiently small for non-positive ones $m < 0$ for all $n_r, \ell$ studied.

The paper studies semiclassical estimates of bound-state energies for $d$-dimensional radial power-law potentials with nonzero angular momentum. It extends an earlier $S$-state prescription using the effective-dimension replacement $d\to d+2\ell$, keeping the centrifugal term out of the action integral. For $V=r^m$, $m>0$, the proposed rule is $\int\sqrt{E-V(r)}\,dr=\pi[n_r+(d+2\ell)/4]$ in units $\hbar=2\mu=1$. For attractive singular powers, the shift is replaced by $(d+2\ell-1)/2$. These choices reproduce the harmonic-oscillator and Coulomb spectra, respectively. Evaluating the action gives explicit beta-function expressions for power-law energy estimates and a logarithmic expression for $V=\log r$. Accuracy is assessed against Lagrange-mesh calculations, principally for $d=2,3$, $\ell=1,\ldots,5$, and radial excitations up to $n_r=100$. For the tested positive powers $m=1,3,4,6$, errors decrease with excitation. A correction $\gamma$, defined from the numerical energies, satisfies $|\gamma_+|<1/2$ in these cases and is reported to decay as $1/n_r$. Two-parameter rational fits for the quartic potential reproduce energies to approximately four significant digits. A quartic comparison finds Langer-corrected quantization more accurate for the tested nonzero angular momenta. The correction is not derived analytically: obtaining exact energies requires numerical input or fitted corrections. The proposed general bound $|\gamma_\pm|\leq1/2$ remains conjectural; logarithmic-potential examples exceed it, and the paper explicitly excludes the tested Coulomb-plus-linear potential from the conjecture.