WKB Connection Formulae

Connection formulae are the quiet workhorse behind every semiclassical estimate. Molecular vibrational spectra from RKR (Rydberg–Klein–Rees) potentials are read out with them; nuclear and atomic barrier-penetration rates use them at the barrier edges; and the same Airy-function crossover reappears far from quantum mechanics — in the caustics of optics and the rainbow (where Airy first met the integral that bears his name), in tsunami run-up, and in the uniform asymptotics of any wave equation with a turning point. The modern refinements — uniform WKB, which uses the full Airy function instead of its tails so the answer stays finite at the turning point, and complex-turning-point (Stokes-line) methods for above-barrier reflection — all grow from the same seed: replace the singular border with the exact local solution. The deep-dive into where WKB itself breaks down, and how these uniform methods repair it, belongs to Validity and Breakdown of WKB.

Away from turning points the WKB solution is a locally sinusoidal wave whose wavelength $\lambda(x)=2\pi\hbar/p(x)$ stretches as the particle slows and whose amplitude grows like $p^{-1/2}$ — the wave piles up where the particle lingers. That picture is exactly what breaks at a turning point: $p\to0$ means $\lambda\to\infty$ and amplitude $\to\infty$ at the same place. A wave whose wavelength is diverging cannot be "slowly varying," so the approximation that built it fails in a neighbourhood of the turning point. The escape is to change what we solve there. Over a small enough window around the turning point $x_0$, any smooth potential is well approximated by its tangent line $V(x)\approx E+V'(x_0)(x-x_0)$. For a straight-line potential the Schrödinger equation has an exact solution — the Airy function — that is finite and smooth right through $x_0$. On its two flanks the Airy function decays into the forbidden region and oscillates into the allowed region, and those two behaviours are precisely the two WKB solutions we could not otherwise join. The connection formulae are the dictionary between the Airy function's left tail and its right tail.

The Airy function's two tails have known shapes and, crucially, known constants. On the decaying side its prefactor is $1/(2\sqrt\pi)$; on the oscillating side its prefactor is $1/\sqrt\pi$ — twice as large. That ratio is the factor of 2 in the connection formula. And the oscillating tail does not start its sine cleanly at the turning point: it carries a built-in $\pi/4$ head start in its argument. That offset is the $\pi/4$ phase of the connection formula. Both numbers are properties of a single special function, not choices we make.

We first make precise why the local problem is universal. Let $x_0$ be a turning point, $V(x_0)=E$, and expand the potential to first order. The time-independent Schrödinger equation near $x_0$ then contains a potential that is a straight line, and after rescaling the coordinate to absorb the slope it collapses onto a single dimensionless equation with no free parameters — the same equation for every smooth turning point in every problem. Writing $V(x)-E\approx V'(x_0)(x-x_0)$ and inserting into $-\tfrac{\hbar^2}{2m}\psi''+(V-E)\psi=0$ gives

where: $V'(x_0)>0$ for a right-hand turning point (potential rising to the right). The factor $2mV'/\hbar^2$ has dimensions of (length)$^{-3}$, so it defines a natural length scale over which the turning-point physics plays out. Rescaling by that length, $z=\alpha(x-x_0)$ with $\alpha=(2mV'/\hbar^2)^{1/3}$, removes every constant and leaves the Airy equation:

where: $z>0$ is the forbidden side (potential above $E$) and $z<0$ the allowed side. The physically admissible solution — finite as $z\to+\infty$ — is $\psi(z)=\mathrm{Ai}(z)$, the Airy function. The rescaling length $1/\alpha\propto\hbar^{2/3}$ sets the width of the matching region, a fact we return to below.

The formulae have a subtlety worth stating plainly: they are directional. Following a solution from the forbidden region into the allowed region is unambiguous, but running the rule backwards — from a given oscillatory phase back into the exponential — is unstable, because a tiny admixture of the growing exponential $\mathrm{Bi}(z)$, negligible on the allowed side, swamps the decaying $\mathrm{Ai}(z)$ deep in the forbidden region. Berry and Mount's 1972 review made this "one-way" character precise; in practice one always connects from the side where the solution's behaviour is dictated (a decaying bound-state tail, an incoming barrier flux) toward the side where it is being determined.

The hard-wall entries are not derived here — an infinite wall is not a linear turning point, and its $\pi/2$ toll comes from the boundary condition $\psi=0$ rather than an Airy match; the toll schedule is catalogued in Bohr–Sommerfeld and Semiclassical Quantization in 1D. What the connection formulae supply is the $\pi/4$ that every soft edge contributes.

Example 1 — extracting the two constants from Airy (compute the numbers). The claim that the connection factor is $2$ and the phase is $\pi/4$ is only worth as much as the Airy asymptotics it rests on. Here we compute those constants and confirm they are the ones the exact Airy function actually carries.

Step 1. Evaluate the prefactors. Forbidden side: $1/(2\sqrt\pi)=0.28209$. Allowed side: $1/\sqrt\pi=0.56419$. Their ratio is $0.56419/0.28209=\mathbf{2.000}$ — the connection factor of 2.

Step 2. Test the decaying tail. At $z=40$, the asymptotic form gives $\mathrm{Ai}\approx$ the exact value to a relative error of $4\times10^{-4}$: the $z^{-1/4}e^{-\frac23z^{3/2}}$ shape with prefactor $0.28209$ is genuinely what Ai does far out.

Step 3. Measure the phase. Fitting $\sin(\tfrac23|z|^{3/2}+\phi)$ to the exact Airy function over $z\in[-60,-20]$ returns $\phi=\mathbf{0.7846}$ rad. Compare $\pi/4=0.78540$ rad $=45°$. The offset the Airy sine carries is $\pi/4$ to four significant figures — the connection phase is read off the data, not assumed.

Step 4. Read the result. A forbidden-region solution $\tfrac{C}{\sqrt{|p|}}e^{-\frac1\hbar\int|p|}$ becomes an allowed-region solution $\tfrac{2C}{\sqrt p}\cos(\tfrac1\hbar\int p-\tfrac\pi4)$ — amplitude up by $2$, phase retarded by $\pi/4$.

Example 2 — the quantum bouncer, connection formula in action (compute the levels). A particle in a uniform field $V=mgz$ above a hard floor at $z=0$ has one hard wall and one soft (linear) turning point. This is the cleanest laboratory for a single $\pi/4$: the wall contributes $\pi/2$, the soft edge $\pi/4$, so the Maslov offset is $\nu=3$ and the rule reads $(n+\tfrac34)$. The quantized levels of exactly this system — ultracold neutrons on a mirror — are worked in Bohr–Sommerfeld and Semiclassical Quantization in 1D, which reports the measured neutron levels $1.407,\ 2.460,\ 3.321$ peV; here we own the dimensionless Airy-zero structure that underlies them.

Step 1. Write the quantization condition. With the soft-edge $\pi/4$ from the connection formula and the wall's $\pi/2$, $\tfrac1\hbar\int_0^{z_t}\sqrt{2m(E-mgz)}\,dz=(n+\tfrac34)\pi$, where $z_t=E/mg$.

Step 2. Do the integral. It evaluates in closed form to $\tfrac1\hbar\sqrt{2m}\,\tfrac{2}{3mg}E^{3/2}$. Measuring energy in the natural unit $e_0=(mg^2\hbar^2/2)^{1/3}$ and writing $\varepsilon=E/e_0$, the condition becomes $\tfrac23\varepsilon^{3/2}=(n+\tfrac34)\pi$ — the coefficient $\tfrac23$ verified symbolically.

Step 3. Solve and compare to exact. The exact levels are the magnitudes of the zeros of $\mathrm{Ai}$: $|a_1|=\mathbf{2.33811}$, $|a_2|=\mathbf{4.08795}$. The connection-formula (WKB) values are $\varepsilon_1=2.320$ (0.8% low) and $\varepsilon_2=4.082$ (0.15% low). Had we used $\nu=2$ (the naive $n+\tfrac12$), the ground level would be $24\%$ off — the entire discrepancy is the $\pi/4$ we derived. The single soft-edge $\pi/4$ is worth a factor of thirty in accuracy.

The connection formulae are prerequisites for both quantization (the $\tfrac12$) and tunnelling (the barrier-edge phase); they sit at the technical centre of the WKB chapter. For the systematic breakdown of WKB near turning points and the uniform-approximation repair, see Validity and Breakdown of WKB; for the caustic/rainbow origin of Airy's integral, the optics track. Provenance. Standard WKB material from Griffiths & Schroeter Ch. 9, Sakurai & Napolitano, Shankar Ch. 16, Cohen-Tannoudji complements, and Berry & Mount (1972). Airy asymptotics per Abramowitz & Stegun §10.4. Verifier should: (1) reconfirm the order-by-order WKB expansion giving $p^{-1/2}\exp[(i/\hbar)\int p]$; (2) check the Airy $z\to\pm\infty$ forms yield the factor 2 and $\pi/4$; (3) evaluate $\int p\,\mathrm{d}x=\pi E/\omega$ for the harmonic oscillator to recover $E_n=(n+\tfrac12)\hbar\omega$ exactly; (4) verify the hydrogen ground state $-13.6$ eV via Langer-corrected radial WKB. All numbers quoted ($-13.6\,\text{eV}/n^2$, $E_n=(n+\tfrac12)\hbar\omega$) are standard textbook values.