Power-like potentials: From the Bohr—Sommerfeld energies to exact ones
Abstract
For one-dimensional power-like potentials |x|m, m>0, the Bohr–Sommerfeld energies (BSE) extracted explicitly from the Bohr–Sommerfeld 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 as opposed to the negative parity states where the BSE remain above the exact ones for m>2 but below them for m<2. The ground state BSE as 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 their relative deviation grows with m, reaching the value 4 at m=∞. For physically important cases m=1,4,6, for the 100th excited state BSE coincide with exact ones in 5–6 figures.
It is demonstrated that by modifying the right-hand side of the Bohr–Sommerfeld quantization condition by introducing the so-called WKB correctionγ (coming from the sum of higher-order WKB terms taken at the exact energies or from the accurate boundary condition at turning points) to the so-called exact WKB condition one can reproduce the exact energies. It is shown that the WKB correction is a small, bounded function |γ|<1/2 for all m≥1. It grows slowly with increasing m for fixed quantum number N, while it decays with quantum number growth at fixed m. It is the first time when for quartic and sextic oscillators the WKB correction and energy spectra (and eigenfunctions) are found in explicit analytic form with a relative accuracy of 10−9–10−11 (and 10−6).
| You currently do not have access to the full text article. |
|---|
