WKB states and Bohr–Sommerfeld quantization

Use P_h=−h²∂ₓ²/2+V and an ansatz u=a exp(iS/h), with real phase in the allowed region. Expanding the derivatives separates the order-one eikonal equation from the order-h transport equation. Writing p(x)=S′(x), the positive branch is sqrt(2(E−V(x))) where E>V. The leading amplitude is a constant times p^(−1/2) on a branch without turning points. It is not constant when the classical particle speed changes.

The approximation is local and requires bounds away from p=0 plus smooth coefficients and an asymptotic remainder construction. Merely solving the first two formal equations does not establish a globally small residual to every order. At a simple turning point V(x)=E with V′ nonzero, the p^(−1/2) amplitude diverges and the oscillatory coordinate chart fails. The exact eigenfunction need not diverge there. An Airy approximation supplies a different local representation that connects oscillatory and decaying regions.

For a precise regular-energy setting, take a smooth real potential equal to a positive quadratic outside a compact set, and a fixed compact energy interval on which exactly two simple turning points bound one allowed well. There are no competing wells at those energies and the turning points stay separated and nondegenerate. The self-adjoint operator has discrete eigenvalues there. Airy matching at both ends gives the closed action rule below, with total Maslov index two and no order-h subprincipal term for this Schrödinger operator.

The closed action is twice the integral of positive momentum between the endpoints. Each simple soft turning point contributes to the phase matching, producing the shift n+1/2 rather than n. Hard boundaries, degenerate turning points and multiple-well matching require different arguments; the same formula cannot simply be copied into them. The O(h²) action remainder is uniform in the specified regular energy window. It is not a uniform assertion at a coalescing pair of turning points as the energy approaches the bottom with h.

For a harmonic well V=ω²x²/2, an energy curve is an ellipse with position radius sqrt(2E)/ω and momentum radius sqrt(2E). Its area is J(E)=2πE/ω, and its period is the action derivative 2π/ω. Inserting this action into the rule gives hω(n+1/2). The exact oscillator spectrum has precisely those values, but its exact solvability is stronger than the general WKB construction and should not be inferred for an anharmonic well.

At a fixed positive energy, the state index grows like one over h. Holding n fixed instead drives oscillator energy toward the minimum and lies in a different asymptotic regime. The harmonic formula remains exact there by direct solution, while a general potential requires a well-bottom harmonic approximation and its corrections. Keeping these regimes distinct prevents an error estimate proved near a noncritical energy from being advertised as a uniform description of every low-lying eigenvalue.