Connection Formulas (Overview)

The WKB wavefunction comes in two flavours — oscillatory where $E > V$ , exponential where $E < V$ — and both diverge exactly at the turning point that separates them, because $p \to 0$ there. The connection formulas tell us how to join the oscillatory solution on one side to the exponential solution on the other, so a single global wavefunction can be assembled.

Why a special treatment is needed

Right at a classical turning point $x_0$ , where $E = V(x_0)$ , the local momentum vanishes, the wavelength diverges, and the WKB amplitude $1/\sqrt{|p|}$ blows up. The slowly-varying assumption fails in a small neighbourhood of $x_0$ . We cannot simply equate the two WKB pieces there; we need a solution valid across the turning point to act as a translator.

The Airy-function patch

Near a simple turning point the potential is approximately linear: $V(x) \approx V(x_0) + V'(x_0)\,(x - x_0)$ . The Schrödinger equation with a linear potential is the Airy equation, whose exact solutions are the Airy functions $\mathrm{Ai}$ and $\mathrm{Bi}$ . The strategy is:

Use the oscillatory WKB form far on the allowed side.
Use the exponential WKB form far on the forbidden side.
In a window around $x_0$ , replace both by the exact Airy solution of the linearized problem.
Match each WKB form to the Airy function's known asymptotics in the overlapping regions.

This matching produces fixed rules relating the amplitudes and phases on the two sides.

The connection rules

For a turning point with the forbidden region to the right, the decaying exponential connects to a phase-shifted cosine on the allowed side:

\frac{1}{\sqrt{|p|}}\exp\!\left(-\frac{1}{\hbar}\int_{x_0}^{x}|p|\,dx\right) \;\;\longleftrightarrow\;\; \frac{2}{\sqrt{p}}\cos\!\left(\frac{1}{\hbar}\int_{x}^{x_0} p\,dx - \frac{\pi}{4}\right).

The crucial output is the $\pi/4$ phase shift picked up at each soft turning point. The growing exponential connects to a sine with the opposite sign of phase and is normally discarded for a bound state (it would blow up deep in the forbidden region).

Where the half-integer comes from

Apply the connection rule at both turning points of a bound state and demand consistency. Each turning point contributes a $\pi/4$ phase deficit, $\pi/2$ in total, and requiring the matched cosines to agree forces

\frac{1}{\hbar}\int_{x_1}^{x_2} p\,dx - \frac{\pi}{2} = n\pi,

which rearranges to the Bohr–Sommerfeld rule

\int_{x_1}^{x_2} p\,dx = \left(n + \tfrac{1}{2}\right)\pi\hbar.

So the mysterious $+\tfrac{1}{2}$ in the quantization condition is not an accident — it is two factors of $\pi/4$ , one from the connection formula at each turning point.

What to take away

Connection formulas are the bookkeeping that makes WKB a complete method rather than two disconnected approximations. They supply the phase shifts that fix the quantization constant and let one stitch a globally valid wavefunction together. Their derivation via the Airy function is technical, but the operational summary — a $\pi/4$ phase per smooth turning point — is all that is needed to apply the quantization rule correctly.