Powerlike potentials: from the BohrSommerfeld energies to exact ones
Abstract
For onedimensional powerlike potentials $x^m, m > 0$ the BohrSommerfeld Energies (BSE) extracted explicitly from the BohrSommerfeld quantization condition are compared with the exact energies. It is shown that for the ground state as well as for all positive parity states the BSE are always above the exact ones contrary to the negative parity states where BSE remain above the exact ones for $m>2$ but they are below them for $m < 2$. The ground state BSE as the function of $m$ are of the same order of magnitude as the exact energies for linear $(m=1)$, quartic $(m=4)$ and sextic $(m=6)$ oscillators but relative deviation grows with $m$ reaching the value 4 at $m=\infty$. For physically important cases $m=1,4,6$ for the $100$th excited state BSE coincide with exact ones in 56 figures. It is demonstrated that modifying the righthandside of the BohrSommerfeld quantization condition by introducing the socalled {\it WKB correction} $\gamma$ (coming from the sum of higher order WKB terms taken at the exact energies) to the socalled exact WKB condition one can reproduce the exact energies. It is shown that the WKB correction is small, bounded function $\gamma < 1/2$ for all $m \geq 1$, it is slow growing with increase in $m$ for fixed quantum number, while it decays with quantum number growth at fixed $m$. For the first time for quartic and sextic oscillators the WKB correction and energy spectra (and eigenfunctions) are written explicitly in closed analytic form with high relative accuracy $10^{9 \ 11}$ (and $10^{6}$).
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2108.00327
 Bibcode:
 2021arXiv210800327D
 Keywords:

 Quantum Physics;
 High Energy Physics  Phenomenology;
 High Energy Physics  Theory
 EPrint:
 16 pages, 4 figures, 3 tables