WKB Bohr-Sommerfeld vs Exact

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.

Figure 1. Bohr-Sommerfeld (WKB) energy levels vs numerical shooting eigenvalues for

V(x) = |x|^p / p

p2.00

n_max6

WHAT TO TRY

Start at p=2, the harmonic oscillator: the WKB and shooting ladders coincide exactly, because Bohr-Sommerfeld is exact for a quadratic potential and the connector lines lie flat.
Slide p up toward 6 so the well steepens toward a box: the two ladders pull apart, most at the ground state. The connector gap is the WKB error, and the readout shows it shrinking from tens of percent at n=0 to a fraction of a percent at the top level.
Raise n_max: more levels appear and the semiclassical trend is direct to read, WKB always converges to the true shooting eigenvalue as n grows, whatever the potential shape.