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.

WKB Bound States in a Smooth Potential applies the two-turning-point rule to the quartic oscillator and measures the resulting spectral error. The derivation of the connection formula remains here.

This page is the canonical home for one-dimensional simple turning-point matching. WKB in Classically Allowed Regions owns the oscillatory branches away from the turning-point layer, while WKB in Classically Forbidden Regions owns the local exponential branches. This page supplies the phase shifts used by Bohr–Sommerfeld Quantization and the matching logic used by Barrier Penetration and Tunneling.

Why WKB Fails

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

The Airy function $\operatorname{Ai}(z)$ decays for $z>0$ and oscillates for $z<0$. Its asymptotic behavior implies the standard decaying-tail connection formula.

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 factor of $2$ and the phase $\pi/4$ are not optional details; they are the content of the connection formula.

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.

Common Mistakes

• Extending $1/\sqrt{p(x)}$ all the way to $p=0$.
• Dropping the factor of $2$ when connecting a decaying forbidden tail to an allowed standing wave.
• Using the same phase correction for smooth turning points and hard walls.
• Treating connection formulas as exact at any barrier thickness. They are asymptotic formulas requiring a region where both the Airy approximation and WKB asymptotics overlap.
• Forgetting orientation. The action integral must run over the correct side of the turning point.

Exercises

1. For $V'(x_0)>0$, identify which side of $x_0$ is classically allowed.

Solution

Near the turning point,

$$V(x)\approx E+V'(x_0)(x-x_0).$$

If $V'(x_0)>0$, then for $x<x_0$ one has $V(x)<E$, so the left side is allowed. For $x>x_0$, $V(x)>E$, so the right side is forbidden.

2. Explain why the exact wavefunction remains finite at a simple turning point even though leading WKB diverges.

Solution

Near a simple turning point the potential is approximately linear, and the Schrödinger equation reduces to the Airy equation. Airy functions are finite at the origin. The WKB divergence comes from using an approximation whose validity condition fails at $p=0$, not from the exact solution.

3. A bound state has two smooth turning points. What is the total turning-point phase contribution to the standing-wave quantization condition?

Solution

Each smooth turning point contributes $\pi/4$ to the allowed-region phase. Two turning points therefore contribute $\pi/2$, which leads to the $n+1/2$ shift in the one-dimensional WKB quantization condition.

References

• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Butterworth-Heinemann, 1981.
• 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.
• F. W. J. Olver, Asymptotics and Special Functions, A K Peters, 1997.