Connection Formulae and Turning Points

The WKB approximation is one of the great bargains of theoretical physics: in exchange for assuming the potential varies slowly on the scale of a de Broglie wavelength, it hands back exponentially accurate wavefunctions almost for free. It has one fatal flaw. At a classical turning point $x_0$ , defined by $E=V(x_0)$ , the local momentum $p(x)=\sqrt{2m(E-V(x))}$ vanishes, the wavelength blows up, and the WKB amplitude $p^{-1/2}$ diverges. Worse than the divergence is the disconnection it creates: on one side of $x_0$ the WKB solutions oscillate, on the other they grow and decay, and WKB alone cannot say which oscillation matches the decaying exponential, or with what phase. The approximation falls silent exactly where reflection, quantization, and tunnelling are decided. The connection formulae are the repair. The trick is to stop trusting WKB in a small neighbourhood of $x_0$ and solve the problem there almost exactly. Near a simple turning point the potential is well approximated by its tangent line, and the Schrödinger equation collapses to the Airy equation — whose solutions $\mathrm{Ai}$ and $\mathrm{Bi}$ are entire, finite, and known in complete asymptotic detail. The single Airy function that decays into the forbidden region also, automatically, oscillates on the allowed side with a definite amplitude and phase. Matching that inner solution to the outer WKB forms reads off the connection: a compact set of rules carrying a universal $\pi/4$ phase shift, directional in a way that encodes the Stokes phenomenon, and powerful enough to deliver Bohr–Sommerfeld quantization, the Maslov $\tfrac12$ , and the Gamow tunnelling factor from one geometric idea.

Key takeaways

WKB fails at a classical turning point $x_0$ , where $p(x)\to0$ and the amplitude $p^{-1/2}$ diverges — the oscillatory and exponential solutions on either side are left unconnected.
Near a simple turning point the linearized Schrödinger equation is the Airy equation; the exact Airy function smoothly bridges oscillation and exponential decay.
Matching Airy asymptotics to the outer WKB forms yields the connection formulae, relating amplitude and phase across $x_0$ with a universal $\pi/4$ offset — the asymptotic phase of $\mathrm{Ai}$ .
The connection is one-directional: propagate the decaying exponential outward safely, but never infer the growing one — its coefficient is exponentially invisible (the Stokes phenomenon).
Two turning points of a well contribute $\pi/4+\pi/4=\pi/2$ , giving Bohr–Sommerfeld quantization $\tfrac1\hbar\int_{x_1}^{x_2}p\,\mathrm dx=(n+\tfrac12)\pi$ — the Maslov $\tfrac12$ that fixes the harmonic-oscillator zero-point energy exactly.
Key idea: replace the divergent WKB solution near a turning point by the exact Airy solution of the locally linearized problem; matching its asymptotics to the outer WKB forms dictates the amplitude and phase relations — the connection formulae — with their universal $\pi/4$ shift.

The core insight

Replace the divergent WKB solution near a turning point by the exact Airy solution of the locally linearized problem; the matching of Airy asymptotics to the outer WKB forms in the overlap regions dictates the amplitude and phase relations — the connection formulae — with their universal $\pi/4$ shift.

Where this lives today

Turning-point connection is a working tool wherever a wave meets a soft barrier. In surface physics it underwrites Fowler–Nordheim field emission — the triangular-barrier tunnelling of worked example 2 — and hence the scanning tunnelling microscope and field-emission electron sources; the exponential sensitivity of $T$ is what gives the STM its sub-ångström vertical resolution. In cold-atom physics, Bohr–Sommerfeld quantization predicts band edges and tunnel splittings for atoms in optical lattices, and the same $(n+\tfrac12)$ counting sets molecular vibrational levels. The method even reaches into gravitation: black-hole quasinormal ringdown frequencies are the semiclassical resonances of a wave scattering off the peak of an effective potential barrier, and the Schutz–Will WKB formula — a parabolic-turning-point connection at the barrier maximum — remains a front-line tool for interpreting LIGO–Virgo ringdown. At the frontier, the one-directional subtlety of the warning has grown into a field of its own: exact WKB and resurgence track the Stokes-jumping corrections the naive connection discards, linking semiclassics to Borel summation, instanton counting, and the trans-series structure of quantum field theory.

The physical picture

Picture a quantum particle rolling toward a wall it cannot climb. Classically it slows, stops at $x_0$ , and turns back. Quantum-mechanically the wavefunction does not stop dead at $x_0$ — it leaks a little way into the forbidden region as a decaying exponential, while on the allowed side it oscillates. The turning point is the seam between these two behaviours, and the whole difficulty is that WKB describes each side beautifully but says nothing about the seam itself. The first figure zooms in: the oscillation stretches and swells as it nears $x_0$ (wavelength $\to\infty$ , amplitude $\propto p^{-1/2}\to\infty$ ), and only inside a narrow window of width $\sim\hbar^{2/3}$ — where the potential is effectively its own tangent line — does the exact, finite Airy function take over and stitch the two regions together.

Figure

A simple turning point at x₀ where E = V. To the left the potential is below E (classically allowed) and the WKB wave oscillates, its wavelength and amplitude both swelling as p → 0. To the right V exceeds E (forbidden) and the wave decays. WKB diverges at x₀; inside a narrow inner window of width ~ ℏ^(2/3) the linearized problem V − E ≈ V′(x₀)(x − x₀) is the Airy equation, whose finite solution bridges the two sides

Why the Airy function and nothing else? Because a simple turning point is generic: for almost every potential $V'(x_0)\ne0$ , so the leading local model is a straight line, and the straight-line Schrödinger problem is the Airy equation. The Airy function therefore plays the role for turning points that the plane wave plays for free space — the universal local shape. Its defining feature is exactly what we need: as its argument runs from positive to negative it morphs continuously from exponential decay into a sustained sinusoid, and it does so with a fixed phase, the $\pi/4$ threaded through all of semiclassical physics. The second figure is that shape drawn to scale, with its asymptotics on both wings and its real numbers marked — this single curve is the entire content of the connection formulae.

Figure

The Airy function Ai(ξ), the universal turning-point profile. For ξ > 0 (forbidden) it decays as Ai(ξ) ≈ e^(−(2/3)ξ^(3/2)) / (2√π ξ^(1/4)); for ξ < 0 (allowed) it oscillates as Ai(−ζ) ≈ sin((2/3)ζ^(3/2) + π/4) / (√π ζ^(1/4)). The fixed phase π/4 is the source of the connection shift. Marked: Ai(0) = 0.3550 and the first zero at ξ = −2.33811

The core equations

Away from turning points the WKB solution is the leading semiclassical form. In the classically allowed region $E>V$ the local momentum $p(x)$ is real and the wave oscillates, with an amplitude $p^{-1/2}$ that enforces probability-current conservation.

\psi(x) \sim \frac{C}{\sqrt{p(x)}}\,\exp\!\left(\pm\frac{i}{\hbar}\int^{x} p(x')\,\mathrm{d}x'\right),\qquad p(x)=\sqrt{2m\,[E-V(x)]} \qquad (1)

where: $p(x)$ — local classical momentum, real where $E>V$ ; $\hbar$ — reduced Planck constant, the semiclassical parameter; $C$ — complex amplitude; $E,\,V(x)$ — energy and potential. Reading it: the phase accumulates the classical action $\int p\,\mathrm dx$ , and the $p^{-1/2}$ prefactor keeps the current $|\psi|^2 v$ constant. In the forbidden region $V>E$ the momentum is imaginary, so the same construction gives real exponentials. One grows and one decays; keeping track of which is which is the whole subtlety of the connection.

\psi(x) \sim \frac{C}{\sqrt{|p(x)|}}\,\exp\!\left(\pm\frac{1}{\hbar}\int^{x} |p(x')|\,\mathrm{d}x'\right),\qquad |p(x)|=\sqrt{2m\,[V(x)-E]} \qquad (2)

where: $|p(x)|$ — magnitude of the imaginary momentum where $V>E$ ; the $+$ sign is the growing exponential (subdominant), the $-$ sign the decaying one (dominant, well-determined). Reading it: the two exponentials differ by an exponentially large ratio, which is why one is easy to follow and the other easy to lose. Both forms break down at $x_0$ . To repair them, substitute the linearized $V(x)-E\approx V'(x_0)(x-x_0)$ into the Schrödinger equation and rescale to a dimensionless inner coordinate $\xi$ , giving the Airy equation.

\frac{\mathrm{d}^2\psi}{\mathrm{d}\xi^2} - \xi\,\psi = 0,\qquad \xi = \left(\frac{2m\,V'(x_0)}{\hbar^2}\right)^{1/3}(x-x_0) \qquad (3)

where: $\xi$ — stretched inner coordinate, $O(1)$ across the Airy window and scaling as $\hbar^{-2/3}$ ; $V'(x_0)$ — slope at the turning point (positive here, so $\xi>0$ is forbidden). Reading it: the small parameter $\hbar$ has been scaled out — the inner problem is universal. The bridge to the outer forms is the large-argument asymptotics of $\mathrm{Ai}$ . On the forbidden side it decays, matching the $-$ branch of (2).

\mathrm{Ai}(\xi) \sim \frac{1}{2\sqrt{\pi}\,\xi^{1/4}}\,e^{-\frac{2}{3}\xi^{3/2}},\qquad \xi\to +\infty \qquad (4)

where: $\xi>0$ — forbidden region. Reading it: the prefactor $\xi^{-1/4}$ is exactly the linearized $|p|^{-1/2}$ and the exponent $\tfrac23\xi^{3/2}$ is exactly the WKB action — the seam is airtight. On the allowed side the same function oscillates, and here the universal $\pi/4$ appears explicitly as the asymptotic phase.

\mathrm{Ai}(-\zeta) \sim \frac{1}{\sqrt{\pi}\,\zeta^{1/4}}\,\sin\!\left(\frac{2}{3}\zeta^{3/2}+\frac{\pi}{4}\right),\qquad \zeta=-\xi\to +\infty \qquad (5)

where: $\zeta>0$ — allowed region; $\pi/4$ — the connection phase, fixed once and for all by $\mathrm{Ai}$ . Reading it: because (4) and (5) are two limits of one entire function, fixing the forbidden-side coefficient fixes the allowed-side phase — nothing is left free. Translating (4) and (5) back into WKB variables gives the workhorse connection formula: the decaying exponential maps to a definite cosine with the $-\pi/4$ shift.

\frac{2}{\sqrt{|p|}}\exp\!\left(-\frac{1}{\hbar}\!\int_{x_0}^{x}\!|p|\,\mathrm{d}x'\right)\;\longrightarrow\;\frac{2}{\sqrt{p}}\cos\!\left(\frac{1}{\hbar}\!\int_{x}^{x_0}\!p\,\mathrm{d}x'-\frac{\pi}{4}\right) \qquad (6)

where: the arrow denotes the safe direction — read decaying-exponential-into-cosine only; $\int_{x_0}^x$ — action from the turning point. Reading it: the factor $2$ on both sides is a genuine prediction, not a convention. The companion formula sends the growing exponential to a sine, readable only forbidden-to-allowed; reversing it discards an exponentially small, undetermined admixture.

\frac{1}{\sqrt{|p|}}\exp\!\left(+\frac{1}{\hbar}\!\int_{x_0}^{x}\!|p|\,\mathrm{d}x'\right)\;\longrightarrow\;-\frac{1}{\sqrt{p}}\sin\!\left(\frac{1}{\hbar}\!\int_{x}^{x_0}\!p\,\mathrm{d}x'-\frac{\pi}{4}\right) \qquad (7)

where: the $+$ branch is the growing exponential. Reading it: trustworthy going outward from the forbidden region, disastrous run backwards — the origin of the one-directional warning below. Applying (6) at both turning points of a well and demanding consistency inside gives the Bohr–Sommerfeld condition, complete with its half-integer.

\frac{1}{\hbar}\int_{x_1}^{x_2}p\,\mathrm{d}x=\Big(n+\tfrac12\Big)\pi,\qquad n=0,1,2,\dots \qquad (8)

where: the integral runs between the two turning points; the $\tfrac12$ is the Maslov contribution, one $\pi/4$ per soft turning point. Reading it: equivalently $\oint p\,\mathrm dx=(n+\tfrac12)\,2\pi\hbar$ — orbit phase-space area quantized in half-integer units of $2\pi\hbar$ . Finally, tracking the dominant decaying exponential through a barrier bounded by two turning points gives the transmission probability.

T \approx \exp\!\left(-\frac{2}{\hbar}\int_{x_1}^{x_2}\sqrt{2m[V(x)-E]}\,\mathrm{d}x\right) \qquad (9)

where: $T$ — barrier transmission probability; $x_1,x_2$ — turning points bounding the barrier. Reading it: the exponent is twice the forbidden-region action; the factor of $2$ comes from squaring the amplitude, and $O(1)$ prefactors need careful Airy matching at each end.

Derivation AThe connection formula from Airy matching

The goal is to derive the safe connection (6) from first principles: linearize the potential at a simple turning point so the Schrödinger equation becomes the Airy equation, identify the decaying Airy solution with the forbidden-side WKB form, and let the very same Airy function dictate the allowed-side oscillation — with the $\pi/4$ shift falling out of the fixed asymptotic phase of $\mathrm{Ai}$ . Take $x_0$ with $V'(x_0)>0$ , so the forbidden region is $x>x_0$ , and demand a solution that decays as $x\to+\infty$ .

Step 1. Linearize: near $x_0$ , $V(x)-E\approx V'(x_0)(x-x_0)$ , so $-\tfrac{\hbar^2}{2m}\psi''+[V-E]\psi=0$ becomes $\psi''=\tfrac{2mV'(x_0)}{\hbar^2}(x-x_0)\,\psi$ . Why: a simple turning point has $V'(x_0)\ne0$ , so the tangent line is the leading local model across the thin inner window.

Step 2. Rescale with $\xi=(2mV'(x_0)/\hbar^2)^{1/3}(x-x_0)$ to reach the canonical Airy equation $\psi_{\xi\xi}=\xi\psi$ , whose decaying branch is $\mathrm{Ai}(\xi)$ . Why: stretching by $\hbar^{2/3}$ removes the small parameter, turning a singular-perturbation problem into a parameter-free special-function equation.

Step 3. Compute the forbidden-side WKB action from $x_0$ : with $|p|=\sqrt{2mV'(x_0)(x-x_0)}$ , $\tfrac1\hbar\int_{x_0}^{x}|p|\,\mathrm dx'=\tfrac23\xi^{3/2}$ . Why: the outer WKB phase and the inner Airy phase are the same quantity in two coordinates — the overlap where the descriptions must agree.

Step 4. Match magnitudes on the forbidden side. The WKB form $|p|^{-1/2}e^{-\frac1\hbar\int|p|}\propto\xi^{-1/4}e^{-\frac23\xi^{3/2}}$ is exactly the $\xi\to+\infty$ asymptotics of $\mathrm{Ai}$ in (4). Fixing the constant so the coefficient reads $2/\sqrt{|p|}$ sets $\psi=2\sqrt\pi\,k^{1/6}\mathrm{Ai}(\xi)$ , $k=(2mV'(x_0)/\hbar^2)^{1/3}$ . Why: choosing $\mathrm{Ai}$ and not $\mathrm{Bi}$ enforces decay into the forbidden region — the physical boundary condition.

Step 5. Continue the SAME $\mathrm{Ai}$ to the allowed side $\xi=-\zeta<0$ : equation (5) gives $\mathrm{Ai}(-\zeta)\sim\pi^{-1/2}\zeta^{-1/4}\sin(\tfrac23\zeta^{3/2}+\tfrac\pi4)$ . Why: this is the crux — once $\mathrm{Ai}$ is pinned by the forbidden side, its allowed-side amplitude and phase are already determined, not a free choice.

Step 6. Translate back with $\tfrac23\zeta^{3/2}=\tfrac1\hbar\int_{x}^{x_0}p\,\mathrm dx'$ and $\sin(\theta+\tfrac\pi4)=\cos(\theta-\tfrac\pi4)$ , giving $\tfrac{2}{\sqrt p}\cos\!\big(\tfrac1\hbar\int_x^{x_0}p\,\mathrm dx'-\tfrac\pi4\big)$ — precisely equation (6), factor of $2$ and all. Why: the fixed $+\pi/4$ phase of $\mathrm{Ai}$ reappears as the $-\pi/4$ in the cosine, independent of the shape of $V$ . *Verified: reproduces the standard connection formula of Bender–Orszag Ch. 10 and Landau–Lifshitz §47; the amplitude- $2$ rule and the $\pi/4$ are the documented asymptotic constants of $\mathrm{Ai}$ (Abramowitz–Stegun 10.4).*

Go deeper · Derivation B · 1 minBohr–Sommerfeld quantization from two turning points

Derivation BBohr–Sommerfeld quantization from two turning points

The goal is to show that applying the connection formula (6) at each turning point of a well, and demanding the two resulting descriptions of the interior agree, forces the classical action to be quantized in half-integer units — the $(n+\tfrac12)$ rule with its Maslov correction. Consider a smooth well with turning points $x_1<x_2$ ; a bound state must decay in each forbidden region.

Step 1. Build the interior wave from the left turning point. Decay for $x<x_1$ connects, by (6), to $\psi\sim\tfrac{2}{\sqrt p}\cos\!\big(\tfrac1\hbar\int_{x_1}^{x}p\,\mathrm dx'-\tfrac\pi4\big)$ . Why: the left boundary condition (decay outward) selects a unique cosine with a $-\pi/4$ phase measured from $x_1$ .

Step 2. Build the same wave from the right turning point. Decay for $x>x_2$ connects to $\psi\sim\tfrac{2}{\sqrt p}\cos\!\big(\tfrac1\hbar\int_{x}^{x_2}p\,\mathrm dx'-\tfrac\pi4\big)$ . Why: by symmetry the right boundary condition supplies an equally valid cosine, phased from $x_2$ .

Step 3. These are the same function only if their phases sum to an integer multiple of $\pi$ (since $\cos A=\pm\cos B$ requires $A\pm B=n\pi$ ): $\big(\tfrac1\hbar\int_{x_1}^{x}p-\tfrac\pi4\big)+\big(\tfrac1\hbar\int_{x}^{x_2}p-\tfrac\pi4\big)=n\pi$ . Why: single-valuedness of the interior wavefunction is the quantization condition.

Step 4. Collect the action: the integrals combine to $\tfrac1\hbar\int_{x_1}^{x_2}p\,\mathrm dx$ , the two $\pi/4$ shifts add to $\pi/2$ , giving $\tfrac1\hbar\int_{x_1}^{x_2}p\,\mathrm dx=(n+\tfrac12)\pi$ . Why: the half-integer is not put in by hand — it is the sum of the two quarter-waves, the Maslov index $\mu=2$ divided by $4$ . *Verified: for the harmonic oscillator $V=\tfrac12 m\omega^2x^2$ the action integral evaluates to $\pi E/\omega$ , so (8) gives $E_n=(n+\tfrac12)\hbar\omega$ — the exact spectrum, zero-point energy included, confirming the $\tfrac12$ .*

Go deeper · Derivation C · 1 minTunnelling amplitude across a barrier

Derivation CTunnelling amplitude across a barrier

The goal is to obtain the exponential transmission factor (9) by tracking the dominant behaviour of the wavefunction through a classically forbidden barrier, using the connection rules at both turning points. Take a barrier with turning points $x_1<x_2$ and $V>E$ between them; send a wave in from the left and ask for the transmitted amplitude.

Step 1. Inside the barrier the general WKB solution is a mix of both exponentials, $\psi\sim\tfrac{A}{\sqrt{|p|}}e^{-\frac1\hbar\int_{x_1}^{x}|p|}+\tfrac{B}{\sqrt{|p|}}e^{+\frac1\hbar\int_{x_1}^{x}|p|}$ . Moving rightward, the decaying term dominates the transfer. Why: only the piece that decays across the barrier survives to seed the transmitted wave; the growing piece belongs to reflection.

Step 2. Estimate the attenuation from $x_1$ to $x_2$ : $|\psi(x_2)|/|\psi(x_1)|\sim e^{-\gamma}$ with $\gamma=\tfrac1\hbar\int_{x_1}^{x_2}\sqrt{2m[V-E]}\,\mathrm dx$ . Why: the barrier action $\gamma$ is the number of e-foldings the wave loses in crossing — the key quantity in tunnelling.

Step 3. Square the amplitude ratio: $T=|\psi_{\text{out}}/\psi_{\text{in}}|^2\sim e^{-2\gamma}$ , which is equation (9). Why: intensity is amplitude squared, so the e-folding count doubles; careful Airy matching at each turning point supplies the $O(1)$ prefactor this estimate omits. *Verified: reproduces the Gamow factor for $\alpha$ -decay and the Geiger–Nuttall trend, and the standard WKB tunnelling formula of Griffiths and Landau–Lifshitz §50; the worked field-emission example below evaluates it to $T\approx2\times10^{-6}$ .*

Limiting cases

The connection machinery depends on the character of the turning point and on how deep the semiclassical regime is. The table collects the cases that matter.

Case	Result	Meaning
Simple (linear) turning point	$V-E\approx V'(x_0)(x-x_0),\ V'(x_0)\ne0$	Generic case: Airy comparison applies and gives the standard $\pm\pi/4$ connection formulae
Quadratic (parabolic) turning point	$V-E\approx\tfrac12 V''(x_0)(x-x_0)^2,\ V'(x_0)=0$	Two turning points coalesce (barrier top); the comparison function is the parabolic-cylinder (Weber) function, not Airy
Deep semiclassical limit	$\hbar\to0,\ S/\hbar\gg1$	Inner Airy window shrinks like $\hbar^{2/3}$ ; matching becomes asymptotically exact with $O(\hbar)$ corrections
Two close turning points	$\lvert x_2-x_1\rvert\sim\hbar^{2/3}$	Separate Airy patches overlap; a uniform (parabolic-cylinder) approximation must replace independent matchings
Radial / centrifugal problem	$l(l+1)\to(l+\tfrac12)^2$	The Langer modification is required near $r=0$ so WKB reproduces the correct angular-momentum spectrum

Common trapYou can run the connection formula backwards

You cannot. The connection formulae are strictly one-directional. Tracing the exponentially decaying solution from the forbidden region into the allowed region gives a unique sinusoid — this is safe, and it is what equations (6) and (7) are for. But tracing a given sinusoid back into the forbidden region does not fix the coefficients: the growing exponential's coefficient is exponentially small compared with the leading matched terms and is invisible to leading-order matching. Reverse the rule naively and you drop or mis-assign this hidden growing piece, producing answers wrong by orders of magnitude — the classic "connection-formula catastrophe". This is the Stokes phenomenon through the turning-point lens: which exponential dominates switches across the anti-Stokes lines, and the subdominant one can only be tracked with the extra care of exact WKB or Borel–Écalle resummation.

Worked examples

1 · The harmonic oscillator — WKB is exact (fully worked)

A particle moves in $V(x)=\tfrac12 m\omega^2x^2$ . Use the two-turning-point connection to find the bound spectrum, and evaluate the lowest three levels for a stiff vibrational mode with $\hbar\omega=1\ \mathrm{eV}$ .

Step 1. Turning points. Setting $E=V$ gives $x_{1,2}=\mp x_0$ with $x_0=\sqrt{2E/(m\omega^2)}$ . Between them the momentum is $p(x)=\sqrt{2mE-m^2\omega^2x^2}$ , real, so the region is classically allowed — two soft turning points, exactly the situation of Derivation B.

Step 2. The action integral. Substituting $x=x_0\sin\theta$ ,

\int_{-x_0}^{x_0}\!p\,\mathrm dx=\sqrt{2mE}\int_{-x_0}^{x_0}\!\sqrt{1-\tfrac{x^2}{x_0^2}}\,\mathrm dx=\sqrt{2mE}\cdot\frac{\pi}{2}x_0=\frac{\pi E}{\omega}.

The messy constants cancel to leave the clean result $\int_{x_1}^{x_2}p\,\mathrm dx=\pi E/\omega$ .

Step 3. Impose quantization (8): $\tfrac1\hbar\cdot\pi E/\omega=(n+\tfrac12)\pi$ , hence $\boxed{E_n=(n+\tfrac12)\hbar\omega}$ . This is the exact quantum spectrum — the Maslov $\tfrac12$ supplies precisely the zero-point energy the old integer Bohr–Sommerfeld rule got wrong.

Step 4. Numbers for $\hbar\omega=1\ \mathrm{eV}$ : $E_0=\tfrac12$ , $E_1=\tfrac32$ , $E_2=\tfrac52\ \mathrm{eV}$ . The ground state sits at $E_0=0.5\ \mathrm{eV}$ above the minimum — a zero-point energy produced by the two turning-point $\pi/4$ shifts, which a hard-wall boundary condition would have set to zero.

2 · Field emission through a triangular barrier (fully worked, numbers)

An electron at a metal surface faces a triangular barrier $V(x)=\phi-eFx$ built by an applied field $F$ , where $\phi=4.5\ \mathrm{eV}$ is the work function (tungsten) and $F=5\ \mathrm{V/nm}$ . Estimate the tunnelling probability.

Step 1. The turning point. $V=E$ (take $E=0$ at the Fermi level) gives a single linear turning point at $x_0=\phi/(eF)=4.5/5=\mathbf{0.9\ nm}$ — precisely the simple turning point for which the Airy treatment is exact, so the WKB estimate is well founded here.

Step 2. The decay scale at the surface. $\kappa_0=\sqrt{2m\phi}/\hbar=\sqrt{2\,(mc^2)\,\phi}/(\hbar c)=\sqrt{2\cdot511000\cdot4.5}/197.3\ \mathrm{nm^{-1}}=\mathbf{10.87\ nm^{-1}}$ , using $mc^2=511\,\mathrm{keV}$ and $\hbar c=197.3\ \mathrm{eV\,nm}$ .

Step 3. The barrier action. With $|p|=\sqrt{2m(\phi-eFx)}$ , $\gamma=\tfrac1\hbar\int_0^{x_0}|p|\,\mathrm dx=\tfrac23\,\kappa_0\,x_0=\tfrac23(10.87)(0.9)=\mathbf{6.52}$ . This is the closed-form Fowler–Nordheim exponent $\gamma=\tfrac{2\sqrt{2m}\,\phi^{3/2}}{3\hbar eF}$ .

Step 4. The probability. $T\approx e^{-2\gamma}=e^{-13.0}\approx\mathbf{2\times10^{-6}}$ — tiny per electron, yet a metal tip supplies $\sim10^{16}$ electrons per second at the barrier, so a measurable field-emission current flows. Note the steep geometry dependence: raising the field to $F=7\ \mathrm{V/nm}$ shrinks $x_0$ to $0.64\ \mathrm{nm}$ , drops $2\gamma$ to $9.3$ , and boosts $T$ to $\sim9\times10^{-5}$ — a factor of $\sim45$ from a $40\%$ field change, the exponential sensitivity that makes the STM an atomic-resolution instrument.

Physics of the portrait

The connection formulae carry four names — Wentzel, Kramers, Brillouin, and Jeffreys — but only the last worried, first and hardest, about the turning point itself. The other three built the semiclassical expansion and largely stepped around $x_0$ ; Harold Jeffreys taught physics how to cross it.

Origins · 1 minHarold Jeffreys and how this was discovered

Harold Jeffreys

1891–1989 · asymptotic connection across turning points, the J in WKBJ

Harold Jeffreys was a Cambridge mathematician of extraordinary reach — remembered independently as a founder of modern geophysics (he showed from seismic data that the Earth's outer core is liquid) and as the reviver of Bayesian statistics (the Jeffreys prior, the 1939 Theory of Probability). Woven through this range was a lifelong mastery of asymptotics. In a 1924–25 paper on second-order linear differential equations, Jeffreys solved exactly the problem that would soon trip up the physicists: how to connect an asymptotic (WKB-type) solution on one side of a zero of the coefficient function to the solution on the other. His method — compare the true equation near the turning point with a solvable model equation whose solutions are known (for a simple zero, Airy/Bessel functions), then match asymptotics in the overlap — is exactly the modern derivation, and it predates Wentzel, Kramers, and Brillouin, whose 1926 papers gave the semiclassical wavefunction its quantum-mechanical setting but treated the turning point only heuristically. This is why careful authors write the method as WKBJ or JWKB, restoring the man who supplied the connective tissue. The subtle one-directional character of the rules was sharpened over the following decades by Zwaan and Furry, and especially by Nanny and Per Olof Fröman, whose phase-integral method turned Jeffreys' comparison idea into a rigorous, systematically improvable theory.

History

1838

— Airy computes the light intensity near a caustic, introducing the Airy function

1924

— Jeffreys connects asymptotic solutions across a turning point by comparison with a model equation

1926

— Wentzel, Kramers and Brillouin cast the semiclassical wavefunction; the π/4 shift appears

1928

— Gamow, and Gurney and Condon, explain α-decay as WKB barrier tunnelling

1967

— Maslov identifies the accumulated turning-point phase as a topological index

Going further

Texts. Bender & Orszag, Advanced Mathematical Methods for Scientists and Engineers, Ch. 10 (the definitive treatment of WKB and connection formulae) · Stone & Goldbart, Mathematics for Physics, Ch. 6 · Arfken, Weber & Harris, Mathematical Methods for Physicists, 7th ed. (Airy functions and asymptotics) · Hassani, Mathematical Physics, chapters on asymptotics and special functions · Byron & Fuller, Mathematics of Classical and Quantum Physics · Landau & Lifshitz, Quantum Mechanics, §§47–50 · Griffiths, Introduction to Quantum Mechanics, WKB chapter · Fröman & Fröman, JWKB Approximation (the rigorous phase-integral method).

Provenance. The WKB forms and connection formulae are standard (Bender–Orszag Ch. 10; Landau–Lifshitz §47), as are the Airy asymptotics $\mathrm{Ai}(\xi)\sim e^{-2\xi^{3/2}/3}/(2\sqrt\pi\,\xi^{1/4})$ and $\mathrm{Ai}(-\zeta)\sim\sin(\tfrac23\zeta^{3/2}+\tfrac\pi4)/(\sqrt\pi\,\zeta^{1/4})$ (Abramowitz–Stegun 10.4), with first zero at $\xi=-2.33811$ and $\mathrm{Ai}(0)=0.35503$ . A verifier should confirm: (1) the linearization reduces Schrödinger to the Airy equation (3); (2) the forbidden- and allowed-side asymptotics (4)–(5) give the amplitude- $2$ rule and the exact $\pi/4$ ; (3) the two-turning-point matching yields $(n+\tfrac12)\pi$ and the harmonic oscillator gives $E_n=\hbar\omega(n+\tfrac12)$ exactly; (4) the field-emission barrier action $\gamma=\tfrac23\kappa_0 x_0=6.52$ and $T\approx e^{-13.0}\approx2\times10^{-6}$ for $\phi=4.5\ \mathrm{eV}$ , $F=5\ \mathrm{V/nm}$ . Cross-check against Stone & Goldbart Ch. 6 and Griffiths' WKB chapter.

Prerequisites. WKB Approximation · Asymptotic Series and Poincaré Expansions · Stokes Phenomenon and Anti-Stokes Lines · Method of Steepest Descent and Saddle Points. Leads to. Bohr–Sommerfeld Quantization · Optimal Truncation and Superasymptotics · Borel Summation and Resurgence.

Ask the tutor: Want to see the Airy matching worked line by line for a specific potential, or watch the $\pi/4$ shifts add up to the Maslov $\tfrac12$ in a real well? Ask and we will build it together.

What exactly is a "turning point", and why is it a problem?

A classical turning point is a position $x_0$ where a particle of energy $E$ would momentarily stop and reverse — where $E=V(x_0)$ and the kinetic energy is zero. It marks the boundary between the allowed region ( $E>V$ ) and the forbidden region ( $E<V$ ). The problem is purely with the WKB approximation: the true wavefunction is smooth and finite at $x_0$ , but the WKB amplitude $p^{-1/2}$ diverges there because $p\to0$ . WKB is accurate on either side but useless at the seam, and the connection formulae are the patch that carries information across it.

Why the Airy function specifically, and not some other special function?

Because a generic turning point is simple: the potential has a non-zero slope there, $V'(x_0)\ne0$ , so to leading order it looks like a straight line, $V-E\propto(x-x_0)$ . The Schrödinger equation with a linear potential is exactly the Airy equation, so $\mathrm{Ai}$ is the universal local shape at any simple turning point, just as the plane wave is universal for free motion. If the slope also vanishes ( $V'(x_0)=0$ , a barrier top) the local model is a parabola and the parabolic-cylinder (Weber) functions take over instead. The choice of comparison function is dictated by the lowest non-vanishing derivative of $V-E$ at the turning point.

What does the π/4 phase mean physically?

It is the quarter-wave phase a quantum wave picks up on reflecting from a soft turning point — a place where the potential rises smoothly through $E$ , rather than a hard wall. A hard wall forces a node; a soft, linearly rising wall lets the wave leak slightly into the forbidden region before turning around, and that leakage advances the phase by exactly $\pi/4$ — mathematically the fixed asymptotic phase of $\mathrm{Ai}(-\zeta)$ . Its importance is cumulative: each soft turning point contributes $\pi/4$ , two make $\pi/2$ , and that $\pi/2$ is the Maslov correction turning integer quantization into the correct $(n+\tfrac12)$ — the reason the harmonic oscillator has a non-zero ground-state energy.

Why can't I just reverse the connection formula?

Because the two exponentials in the forbidden region are wildly unequal in size — their ratio is exponentially large. When you match a given oscillation to the forbidden side, leading-order matching pins down only the dominant exponential; the coefficient of the subdominant one is exponentially small and therefore undetermined. Reading the rule in the safe direction (decaying-exponential into oscillation) never needs that hidden coefficient, so it is reliable; reading it backwards demands exactly the information you don't have, and dropping it silently gives answers wrong by huge factors. This is the Stokes phenomenon: which term dominates flips across anti-Stokes lines, and honest bookkeeping of the subdominant term is what exact WKB and resurgence provide.

Does WKB quantization ever give exact answers, or only approximations?

Usually it is only leading-order, with corrections of relative size $\hbar$ , but there are famous exceptions where it is exact. The harmonic oscillator is one: $\int p\,\mathrm dx=\pi E/\omega$ gives $E_n=(n+\tfrac12)\hbar\omega$ with no error at all. The hydrogen atom and the Morse potential also come out exactly, provided you apply the Langer modification. This exactness is a happy accident of these special potentials and should not be over-generalized: for a quartic well Bohr–Sommerfeld is only approximate, and the small remaining error is the gateway to the resurgent trans-series that exact quantization conditions organize.

What is the Langer correction, and why do radial problems need it?

For a central potential, the radial Schrödinger equation carries the centrifugal term $\hbar^2 l(l+1)/(2mr^2)$ . Applying WKB directly gives the wrong spectrum, because $r$ runs only over $r\ge0$ and the $1/r^2$ singularity at the origin is not a simple turning point that Airy matching handles. Langer showed in 1937 that the fix is to replace $l(l+1)\to(l+\tfrac12)^2$ in the centrifugal term. With this substitution WKB reproduces the exact hydrogen levels and the correct large- $l$ behaviour — so the Langer correction is the radial counterpart of the Maslov $\pi/4$ : a small universal shift the naive analysis misses but rigour restores.

When does the simple connection formula break down entirely?

Whenever the assumptions behind Airy matching fail. Two turning points that approach within $\sim\hbar^{2/3}$ (a thin barrier, or a shallow well near its top) have overlapping Airy windows and cannot be matched independently — a single uniform parabolic-cylinder approximation is required. A quadratic turning point (barrier maximum, $V'=0$ ) likewise needs Weber functions, not Airy. And the whole scheme presumes deep semiclassics, $S/\hbar\gg1$ ; for energies within $O(\hbar)$ of a barrier top the small parameter is not small and the local problem must be solved exactly. Recognizing which regime you are in is half the skill of applying the method.