Turning Points and Connection Formulas

Leading WKB fails at a classical turning point, where $E=V(x)$ and the local momentum vanishes. Connection formulas repair the approximation by replacing the potential near the turning point with a linear approximation, solving the resulting Airy equation, and matching the Airy asymptotics to WKB forms on both sides.

Required background. Allowed-region WKB and forbidden-region WKB give the branches to match. We consider a smooth, isolated simple turning point with nonzero potential slope.

Failure of WKB at a turning point

The allowed-region WKB form contains

$$\frac{1}{\sqrt{p(x)}}, \qquad p(x)=\sqrt{2m\left(E-V(x)\right)}.$$

At a turning point $x_0$,

$$E=V(x_0), \qquad p(x_0)=0.$$

The amplitude appears to diverge, and the validity condition

$$\left| \hbar \frac{p'}{p^2} \right| \ll 1$$

cannot hold. The divergence is not a physical divergence of the exact wavefunction; it is a failure of the local WKB approximation.

Linearized Potential

Assume a simple turning point:

$$V(x_0)=E, \qquad V'(x_0)\ne0.$$

Near $x_0$,

$$V(x) \approx E+F(x-x_0), \qquad F=V'(x_0).$$

For $F>0$, the region $x<x_0$ is classically allowed and $x>x_0$ is forbidden. The Schrödinger equation becomes

$$- \frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} + F(x-x_0)\psi = 0.$$

With the scaled variable

$$z = \left(\frac{2mF}{\hbar^2}\right)^{1/3} (x-x_0),$$

this is the Airy equation

$$\frac{d^2\psi}{dz^2} - z\psi = 0.$$

The local solutions are linear combinations of $\operatorname{Ai}(z)$ and $\operatorname{Bi}(z)$.

The definitions and asymptotic formulas for these functions are collected in Airy Functions.

Airy Matching

Write $\ell=(\hbar^2/(2mF))^{1/3}$, so $z=(x-x_0)/\ell$. The leading Airy asymptotics for positive $u\gg1$ are

$$\begin{aligned} \operatorname{Ai}(u)&\sim\frac{u^{-1/4}}{2\sqrt\pi}e^{-2u^{3/2}/3},\\ \operatorname{Ai}(-u)&\sim\frac{u^{-1/4}}{\sqrt\pi} \sin\left(\frac23u^{3/2}+\frac\pi4\right). \end{aligned}$$

For the linear potential, $2u^{3/2}/3$ is exactly the dimensionless action integral on either side. The amplitude ratio therefore produces the factor of two below; the phase is fixed by the same Airy solution [Olver; NIST DLMF §9.7].

For an allowed region on the left and forbidden region on the right,

$$\frac{C}{\sqrt{\kappa(x)}} \exp\left( - \frac{1}{\hbar} \int_{x_0}^{x}\kappa(x')\,dx' \right)$$

in the forbidden region $x>x_0$ connects to

$$\frac{2C}{\sqrt{p(x)}} \sin\left[ \frac{1}{\hbar} \int_x^{x_0}p(x')\,dx' + \frac{\pi}{4} \right]$$

in the allowed region $x<x_0$.

Here

$$\kappa(x)=\sqrt{2m\left(V(x)-E\right)}.$$

The independent $\operatorname{Bi}$ solution instead connects a growing exponential to a cosine. With $\Xi=\hbar^{-1}\int_{x_0}^x\kappa\,dx'$ and $\Theta=\hbar^{-1}\int_x^{x_0}p\,dx'$, the leading pair is

$$\frac{D e^{-\Xi}+G e^{\Xi}}{\sqrt\kappa} \quad\longleftrightarrow\quad \frac{2D\sin(\Theta+\pi/4)+G\cos(\Theta+\pi/4)}{\sqrt p}.$$

These coefficients refer to the resolved Airy basis. Observing only a dominant growing asymptotic branch does not determine an exponentially subdominant coefficient $D$; retaining that information is essential when matching tunneling amplitudes.

Matching also needs an overlap region. If the linearization varies appreciably over a length $L$, it requires

$$\ell\ll |x-x_0|\ll L.$$

For example, $|V''(x_0)(x-x_0)|\ll |F|$ controls the first neglected Taylor term. Thus $\ell\ll L$ is a scale-separation condition, not merely the existence of a root $V=E$.

Opposite Orientation

If the forbidden region lies to the left and the allowed region lies to the right, the decaying forbidden solution

$$\frac{C}{\sqrt{\kappa(x)}} \exp\left( - \frac{1}{\hbar} \int_x^{x_0}\kappa(x')\,dx' \right)$$

connects to

$$\frac{2C}{\sqrt{p(x)}} \sin\left[ \frac{1}{\hbar} \int_{x_0}^{x}p(x')\,dx' + \frac{\pi}{4} \right].$$

These formulas assume a single simple turning point and a forbidden-side boundary condition that selects the decaying branch. More general scattering situations may require both Airy solutions and a connection matrix rather than a single decaying-tail formula.

Turning-Point Phase

Each smooth turning point contributes a phase shift of $\pi/4$ to the allowed-region standing wave. A bound state trapped between two smooth turning points therefore receives a total phase contribution of $\pi/2$. This is the origin of the $n+1/2$ in the one-dimensional Bohr-Sommerfeld rule.

Hard-wall boundaries, singular endpoints, and higher-order turning points can carry different phase corrections. The common $1/2$ shift is a result for two ordinary smooth turning points, not a universal rule for every boundary condition.

Exercises

1. Express the allowed-side decaying-tail solution as incoming and reflected traveling waves. Use $\Theta=\hbar^{-1}\int_x^{x_0}p\,dx'$ and identify the reflection amplitude.

Solution

Since $\Theta'=-p/\hbar$, $e^{-i\Theta}$ travels toward the right-hand turning point and $e^{i\Theta}$ travels away. Expansion gives

$$2D\sin(\Theta+\pi/4) =\frac{D}{i}\left(e^{i\pi/4}e^{i\Theta}-e^{-i\pi/4}e^{-i\Theta}\right).$$

The reflected-to-incoming coefficient ratio is $r=-i$, with $|r|=1$. This reflection phase $-\pi/2$ is consistent with the standing-wave offset $\pi/4$; they describe different phase comparisons.

2. How does doubling $\hbar$ change the width of the simple-turning-point layer for fixed $m$ and $F$? What happens as $F\to0$?

Solution

The width grows by $2^{2/3}$. As $F\to0$, $\ell$ grows and the linear approximation loses its overlap with WKB. Coalescing or higher-order turning points require a different uniform model; setting $F=0$ in the Airy formula is invalid.

References

• C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Springer, 1999.
• M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics”, Reports on Progress in Physics 35, 315-397, 1972.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Butterworth-Heinemann, 1981.
• National Institute of Standards and Technology, Digital Library of Mathematical Functions, §9.7: Asymptotic Expansions, Airy-function formulas 9.7.5, 9.7.7, 9.7.9, and 9.7.11 (accessed 2026-10-13).
• F. W. J. Olver, Asymptotics and Special Functions, A K Peters, 1997.