WKB Bohr-Sommerfeld vs exact eigenvalues

What you are seeing: bound-state energies for a particle in a power-law well $V(x) = |x|^p / p$, compared two ways. The blue ladder is the Bohr-Sommerfeld (WKB) approximation

The orange ladder is the "exact" answer: for the harmonic oscillator ($p = 2$) we use the closed form $E_n = n + \tfrac12$; for the quartic anharmonic oscillator ($p = 4$) we use the published numerical eigenvalues from Bender and Wu (1969).

The Bohr-Sommerfeld rule is exact for the harmonic oscillator (a small miracle of the harmonic potential). For the quartic well it's an approximation that is excellent for $n$ large and noticeably wrong for $n = 0$. Slide $p$ between 2 and 6 to watch the BS curve drift away from the analytic harmonic ladder.