WKB Approximation

The WKB approximation is the master tool of asymptotics for linear differential equations whose coefficients vary slowly compared with the local scale on which the solution oscillates or decays. Its prototype is $\epsilon^2 y'' = Q(x)\,y$ with $\epsilon$ a formally small parameter. As $\epsilon\to0$ the solution is rapidly varying — oscillatory where $Q<0$ , exponential where $Q>0$ — so it is not analytic in $\epsilon$ at $\epsilon=0$ , and a naive power series in $\epsilon$ (regular perturbation theory) cannot represent it. The escape is deceptively simple: do not expand the solution, expand its logarithm. Writing $y=e^{S/\epsilon}$ moves the singular fast variation into the exponent and turns a linear second-order ODE into a first-order Riccati equation for the phase $S'$ , which does admit a regular series $S=S_0+\epsilon S_1+\cdots$ . That single move organizes the whole subject. The leading term obeys the eikonal equation $(S_0')^2=Q$ — the ray equation of geometric optics, the Hamilton–Jacobi equation of classical mechanics — and the next term is a transport equation that conserves flux and delivers the universal $Q^{-1/4}$ amplitude. WKB is thus the mathematical engine of the correspondence principle: classical mechanics is the short-wavelength limit of quantum mechanics exactly as geometric optics is the short-wavelength limit of the wave equation, with $\hbar$ playing the role of $\epsilon$ . The method has one sharp failure mode: it breaks down at turning points where $Q=0$ , because the $Q^{-1/4}$ amplitude diverges and the local wavelength becomes infinite. Repairing it there requires an exact local solution — the Airy function — whose asymptotics, matched to the WKB forms on either side, produce the connection formulae. Those formulae carry famous $\pi/4$ phase shifts and factors of two that are not bookkeeping accidents but the fingerprint of the Stokes phenomenon in the complex plane. From them follow Bohr–Sommerfeld quantization, Gamow tunnelling rates, and the entire semiclassical toolkit.

Key takeaways

The WKB (Wentzel–Kramers–Brillouin, and earlier Liouville–Green) method solves $\epsilon^2 y''=Q(x)y$ by writing $y=\exp(S/\epsilon)$ and expanding the phase $S=S_0+\epsilon S_1+\cdots$ in the small parameter — never the solution itself.
Leading order gives the eikonal $(S_0')^2=Q$ , so $S_0=\pm\int\sqrt{Q}\,\mathrm{d}x$ ; first order gives the transport equation and $S_1=-\tfrac14\ln Q$ , the universal $Q^{-1/4}$ amplitude that enforces flux conservation.
Where $Q<0$ the solution oscillates as $(-Q)^{-1/4}e^{\pm i\int\sqrt{-Q}/\epsilon}$ ; where $Q>0$ it grows or decays exponentially — the tunnelling branch.
WKB fails at turning points ( $Q=0$ ); the Airy function is the exact local bridge and yields the connection formulae with their $\pi/4$ phase shifts.
Applied to the Schrödinger equation it gives the semiclassical wavefunction $\psi\sim p^{-1/2}e^{\pm i\int p\,\mathrm{d}x/\hbar}$ and, via two turning points, Bohr–Sommerfeld quantization $\oint p\,\mathrm{d}x=2\pi\hbar(n+\tfrac12)$ .
Key idea: do not expand the solution — expand its logarithm. Writing $y=e^{S/\epsilon}$ turns a linear second-order ODE into a Riccati equation for $S'$ , whose $\epsilon$ -expansion yields the eikonal (phase) at leading order and the transport equation (amplitude) at next order, with the universal $Q^{-1/4}$ prefactor.

The core insight

Do not expand the solution — expand its logarithm. Writing $y=e^{S/\epsilon}$ turns a linear second-order ODE into a Riccati equation for $S'$ , whose expansion in $\epsilon$ yields the eikonal (phase) at leading order and the transport (amplitude) at next order, with the universal $Q^{-1/4}$ prefactor.

Where this lives today

WKB is not a historical curiosity; it is a daily workhorse. In semiclassical and quantum-chemistry codes it supplies fast, accurate energy levels and tunnelling splittings for molecular vibrations, where a full solution over many degrees of freedom is prohibitive. In nuclear and atomic physics the Gamow factor still sets alpha-decay lifetimes and the astrophysical reaction rates that power stellar nucleosynthesis. Scanning tunnelling microscopy reads surface topography through the exponential WKB tunnelling current, resolving individual atoms because that current changes by an order of magnitude per angstrom of gap. In wave physics the eikonal branch underlies seismic ray tracing, ionospheric radio propagation, gravitational-lensing time delays, and gradient-index optical design. And at the frontier, WKB is the entry point to resurgence and exact WKB, where the divergent series, its Stokes jumps, and its exponentially small non-perturbative corrections are resummed exactly — an active program linking quantum mechanics, matrix models, and quantum field theory.

The physical picture

Picture a wave whose local wavelength changes slowly across the medium. Over any single wavelength the surroundings look uniform, so locally the solution is an ordinary plane wave $e^{ik(x)x}$ ; WKB stitches these together while conserving flux. Where the wave slows the crests bunch up and energy piles in, which is precisely why the amplitude scales as $1/\sqrt{k}$ , i.e. $Q^{-1/4}$ . The exponential ansatz loads all the fast variation into the phase $S$ , leaving a well-behaved hierarchy for the successive corrections. The drama is all at the turning point, where the wave "turns around": there the local wavelength diverges, the slowly-varying hypothesis collapses, and the oscillatory and exponential descriptions must be glued. The gluing object is the Airy function, the exact solution of the linearized problem $y''=zy$ . The first figure shows this canonical scene — an oscillation on the allowed side swelling toward the turning point along a $(-Q)^{-1/4}$ envelope, and a monotone exponential decay on the forbidden side, with the Airy function bridging the two and fixing the $\pi/4$ phase.

Figure

The WKB wavefunction through a simple turning point x₀ where Q = 0: on the left, classically allowed with Q < 0, it oscillates along an amplitude envelope ∝ (−Q)^(−1/4) that swells toward x₀; on the right, forbidden with Q > 0, it decays exponentially. The exact Airy function Ai bridges the two and supplies the π/4 phase shift

Follow the phase into the complex plane and a deeper structure appears: the two branches $e^{\pm\int\sqrt Q}$ dominate in different sectors and swap roles across Stokes lines. That exchange — the Stokes phenomenon — forces the exact $\pi/4$ phase and factor-of-two in the connection formulae. Its most celebrated payoff is Bohr–Sommerfeld quantization: applying the connection formula at both turning points of a well forces the enclosed action to a half-integer multiple of $2\pi\hbar$ . The second figure shows this producing the harmonic-oscillator ladder — the exact spectrum, with the extra $\tfrac12$ arising purely from two $\pi/4$ turning-point shifts.

Figure

Bohr–Sommerfeld quantization of the harmonic oscillator V = ½mω²x²: requiring the action ∮ p dx = 2πℏ(n + ½) fixes equally spaced levels E_n = (n + ½)ℏω — the exact spectrum. The ½ is the Maslov correction, one π/4 phase shift at each of the two turning points

The core equations

The whole method rests on one canonical second-order equation. Any linear ODE $y''+P(x)y'+R(x)y=0$ can be brought to this form by removing the first-derivative term (the Liouville transformation), so the single coefficient $Q(x)$ carries all the physics: its sign decides oscillatory versus exponential behaviour, and its zeros are the turning points.

\epsilon^2\,\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} = Q(x)\,y \qquad (1)

where: $\epsilon$ — small parameter measuring the slowness of variation (it is $\hbar$ in quantum mechanics, the inverse wavenumber in optics); $Q(x)$ — slowly varying coefficient whose sign sets the character of the solution; $y(x)$ — the unknown to be approximated. Reading it: when $\epsilon$ is small the right side dominates unless $y$ varies on the scale $\epsilon$ , forcing the fast variation the ansatz captures. The central trick is to expand not $y$ but its logarithm, absorbing the fast dependence into an exponent. Substituting this ansatz converts equation (1), order by order, into a solvable hierarchy.

y(x) = \exp\!\Big(\frac{1}{\epsilon}\sum_{n=0}^{\infty}\epsilon^{n}\,S_n(x)\Big) \qquad (2)

where: $S_0$ — the eikonal or leading phase, of order $1/\epsilon$ in the exponent; $S_1$ — the first correction, which supplies the amplitude; $S_n$ — higher corrections, each smaller by one power of $\epsilon$ when WKB is valid. Reading it: the $1/\epsilon$ prefactor makes $y$ non-analytic at $\epsilon=0$ , so this form does what a Taylor series in $\epsilon$ cannot. At leading order the balance is purely algebraic — the eikonal equation. It is the Hamilton–Jacobi equation of the problem: $S_0'$ is the local momentum, and its integral is the classical action.

(S_0')^2 = Q(x) \;\;\Longrightarrow\;\; S_0(x) = \pm\int^{x}\!\sqrt{Q(x')}\,\mathrm{d}x' \qquad (3)

where: $S_0'$ — the local "momentum" $\pm\sqrt Q$ ; the two signs generate the two independent solutions. Reading it: the phase accumulates $\sqrt Q$ per unit length, oscillating when $Q<0$ (so $\sqrt Q$ is imaginary) and growing or decaying when $Q>0$ . At the next order a first-order linear equation fixes the amplitude. It is a conservation law — the transport equation — and integrating it produces the single most recognizable feature of WKB, the inverse quarter-power prefactor.

2\,S_0'\,S_1' + S_0'' = 0 \;\;\Longrightarrow\;\; S_1 = -\tfrac14\ln Q,\qquad e^{S_1}=Q^{-1/4} \qquad (4)

where: $S_1$ — the amplitude exponent; $Q^{-1/4}$ — the universal prefactor. Reading it: $2S_0'S_1'+S_0''=0$ is exactly $\mathrm{d}(S_0'\,e^{2S_1})/\mathrm{d}x=0$ , the statement that the flux $|y|^2 S_0'$ is conserved — amplitude piles up where the wave slows. Combining the two orders gives the working formula. Both branches are kept, with constants fixed by boundary data or by connection across a turning point.

y(x) \sim \frac{1}{Q^{1/4}}\Big[\,c_+\,e^{\frac{1}{\epsilon}\int^x\!\sqrt{Q}\,\mathrm{d}x'} + c_-\,e^{-\frac{1}{\epsilon}\int^x\!\sqrt{Q}\,\mathrm{d}x'}\Big] \qquad (5)

where: $c_\pm$ — constants set by boundary/connection conditions; $Q^{1/4}$ — the amplitude, which diverges at any turning point $Q=0$ . Reading it: away from turning points this form is accurate to relative order $\epsilon^2$ ; the deepest error is the neglected exponentially small partner across Stokes lines. In quantum mechanics equation (1) is the time-independent Schrödinger equation with $Q=-p^2/\hbar^2$ and $\epsilon=\hbar$ , giving the semiclassical wavefunction directly.

\psi_{\text{WKB}}(x) \sim \frac{A}{\sqrt{p(x)}}\,\exp\!\Big(\pm\frac{i}{\hbar}\int^x\! p(x')\,\mathrm{d}x'\Big),\qquad p=\sqrt{2m\,(E-V)} \qquad (6)

where: $p(x)$ — the classical momentum; $\hbar$ — the small parameter, semiclassical limit $\hbar\to0$ ; $A$ — amplitude constant. Reading it: $|\psi|^2\propto 1/p$ , so the particle is most likely found where it moves slowest — the quantum echo of a classical particle lingering near its turning points. Finally, the expansion is only self-consistent while the first correction is genuinely smaller than the leading phase. That single inequality is the validity condition, and it states that the local wavelength must change little over one wavelength.

\left|\frac{\epsilon\,Q'}{Q^{3/2}}\right| = \Big|\frac{\mathrm{d}\lambda}{\mathrm{d}x}\Big| \ll 1,\qquad \lambda(x)=\frac{2\pi\hbar}{|p(x)|} \qquad (7)

where: $\lambda(x)$ — the local de Broglie wavelength; the middle quantity is $|\epsilon S_1'/S_0'|$ . Reading it: the criterion diverges as $Q\to0$ , which is the precise, quantitative reason WKB must fail at every turning point.

Derivation AThe WKB series and the inverse quarter-power amplitude

The goal is to descend from the exact canonical equation to the two-term leading solution using nothing but the exponential ansatz and matching of powers of the small parameter, seeing where the eikonal phase, the transport equation, and the universal amplitude each come from. We substitute, reduce to a Riccati equation, split it order by order, integrate the first two orders, and reassemble — the residual on back-substitution is order $\epsilon^2$ , which is what "leading WKB" means.

Step 1. Insert $y=e^{S/\epsilon}$ with $S=\sum_n\epsilon^n S_n$ . Differentiating twice gives $y'=\tfrac1\epsilon S'y$ and $y''=(\tfrac1{\epsilon^2}(S')^2+\tfrac1\epsilon S'')y$ ; substituting into $\epsilon^2 y''=Qy$ and dividing by $y/\epsilon^2$ yields the exact Riccati relation $(S')^2+\epsilon S''=Q$ . Why: this is the pivot of the method — a linear second-order ODE has become a first-order equation for $S'$ , and its nonlinearity is harmless because we solve it perturbatively.

Step 2. Collect powers of $\epsilon$ in $(S_0'+\epsilon S_1'+\cdots)^2+\epsilon(S_0''+\cdots)=Q$ . At order $\epsilon^0$ only the leading square survives: $(S_0')^2=Q$ , so $S_0=\pm\int\sqrt Q\,\mathrm{d}x$ . Why: the two signs are the two independent solutions of a second-order equation — later "incoming versus outgoing" or "growing versus decaying".

Step 3. At order $\epsilon^1$ the cross term and the second derivative balance: $2S_0'S_1'+S_0''=0$ , so $S_1'=-S_0''/(2S_0')=-\tfrac12\,\mathrm{d}(\ln S_0')/\mathrm{d}x$ . Integrating, $S_1=-\tfrac12\ln S_0'=-\tfrac12\ln\sqrt Q=-\tfrac14\ln Q$ , hence the amplitude $e^{S_1}=Q^{-1/4}$ . Why: it is a conservation law in disguise — the balance integrates to $S_0'e^{2S_1}=\text{const}$ , i.e. constant flux, so $Q^{-1/4}$ is dictated by physics.

Step 4. Assemble $y=e^{S_0/\epsilon+S_1+O(\epsilon)}=Q^{-1/4}e^{\pm\frac1\epsilon\int\sqrt Q}$ and superpose the two branches with constants $c_\pm$ to obtain equation (5). Why: a second-order linear ODE has a two-parameter general solution, and the two eikonal signs supply exactly those parameters, so nothing is lost at leading order. *Verified: back-substitution into equation (1) leaves a residual of relative order $\epsilon^2$ ; the result reproduces the Liouville–Green form and Bender & Orszag Ch. 10, checked numerically for the harmonic-oscillator and linear potentials.*

Go deeper · Derivation B · 1 minTurning-point matching through the Airy equation

Derivation BTurning-point matching through the Airy equation

The goal is to repair WKB exactly where it breaks — at a simple turning point $x_0$ with $Q(x_0)=0$ and $Q'(x_0)>0$ — by replacing the divergent WKB forms with the exact local solution and matching. We linearize the coefficient near the zero, rescale into the universal Airy form, quote the two asymptotic tails of the Airy function, and identify them with the WKB solutions on either side; the identification is forced, and it pins down the $\pi/4$ phase that no purely local WKB argument can supply.

Step 1. Taylor-expand near the simple zero: $Q(x)\approx Q'(x_0)(x-x_0)$ , so equation (1) becomes $\epsilon^2 y''\approx Q'(x_0)(x-x_0)\,y$ . Why: at a simple zero the linear term is the leading behaviour, and it is precisely this linear coefficient that makes the local problem exactly solvable.

Step 2. Rescale with $z=(x-x_0)/\ell$ and $\ell=(\epsilon^2/Q'(x_0))^{1/3}$ to reach the Airy equation $y''=zy$ , whose exact solution is $y=a\,\mathrm{Ai}(z)+b\,\mathrm{Bi}(z)$ . Why: the width $\ell$ is the boundary layer over which WKB is invalid — it shrinks like $\epsilon^{2/3}$ , so the repair region is thin and WKB is recovered outside it.

Step 3. Use the Airy asymptotics: $\mathrm{Ai}(z)\sim \dfrac{e^{-\frac23 z^{3/2}}}{2\sqrt\pi\,z^{1/4}}$ as $z\to+\infty$ (decaying, forbidden side) and $\mathrm{Ai}(z)\sim\dfrac{1}{\sqrt\pi\,|z|^{1/4}}\sin\!\big(\tfrac23|z|^{3/2}+\tfrac\pi4\big)$ as $z\to-\infty$ (oscillatory, allowed side). Why: the same solution decays on one side and oscillates on the other with a built-in $\pi/4$ offset — invisible unless one solves the connection exactly.

Step 4. Identify $\tfrac23 z^{3/2}=\tfrac1\hbar\int_{x_0}^x|p|\,\mathrm{d}x'$ and read off the boxed connection formula $\dfrac{2}{\sqrt{|p|}}\cos\!\big(\tfrac1\hbar\int_x^{x_0}p\,\mathrm{d}x'-\tfrac\pi4\big)\longleftrightarrow \dfrac{1}{\sqrt{|p|}}e^{-\frac1\hbar\int_{x_0}^x|p|\,\mathrm{d}x'}$ . Why: the factor of two and the $-\pi/4$ are not adjustable — they are the numerical content of the Airy asymptotics, and mishandling them corrupts every downstream quantization. *Verified: matches Bender & Orszag §10.4 and Abramowitz & Stegun §10.4; only the decaying-to-cosine direction is well-conditioned, a manifestation of the Stokes phenomenon.*

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

Derivation CBohr–Sommerfeld quantization from two turning points

The goal is to turn the connection formula into a spectrum. For energy $E$ in a smooth well with classical turning points $a<b$ , the interior is allowed and the exterior forbidden; demanding a decaying-on-both-sides wavefunction forces the enclosed action to be quantized. We connect from each wall inward and require the two interior cosines to describe one and the same standing wave — a consistency condition that is the quantization rule, complete with its half-integer Maslov shift.

Step 1. At the right turning point $b$ , the solution that decays for $x>b$ connects inward to $\psi\propto\dfrac{2}{\sqrt p}\cos\!\big(\tfrac1\hbar\int_x^b p\,\mathrm{d}x'-\tfrac\pi4\big)$ . Why: normalizability forbids the growing exponential outside the well, so the exterior branch is fixed and the interior phase follows uniquely.

Step 2. At the left turning point $a$ , symmetrically, the decaying exterior solution connects to $\psi\propto\dfrac{2}{\sqrt p}\cos\!\big(\tfrac1\hbar\int_a^x p\,\mathrm{d}x'-\tfrac\pi4\big)$ . Why: the same normalizability argument applies at the other wall, and each wall contributes its own independent $-\pi/4$ .

Step 3. These two expressions must be the same interior function. Writing the total phase $\tfrac1\hbar\int_a^b p\,\mathrm{d}x$ as the sum of the two partial phases, the cosines agree only if the phases differ by a multiple of $\pi$ : $\tfrac1\hbar\int_a^b p\,\mathrm{d}x-\tfrac\pi2=(n+\ldots)\pi$ . Why: the two accumulated $\pi/4$ shifts are what produce the crucial $-\pi/2$ .

Step 4. Rearranging gives the quantization condition $\tfrac1\hbar\int_a^b p\,\mathrm{d}x=(n+\tfrac12)\pi$ , equivalently $\oint p\,\mathrm{d}x=2\pi\hbar\,(n+\tfrac12)$ , $n=0,1,2,\dots$ . Why: the $+\tfrac12$ is the Maslov correction (Maslov index 2, one $\pi/4$ per soft turning point); a hard wall would contribute $\pi/2$ instead and change the constant. *Verified: for $V=\tfrac12 m\omega^2x^2$ the action integral is $\pi E/\omega$ , so $(n+\tfrac12)\pi\hbar=\pi E/\omega$ gives the exact spectrum $E_n=(n+\tfrac12)\hbar\omega$ , reproduced numerically to 5 digits for $n=0,\dots,3$ .*

Limiting cases

The entire behaviour of the WKB solution is dictated by the sign and the zeros of $Q(x)$ .

Case	Result	Meaning
Classically allowed, $Q<0$	$y\sim \dfrac{C}{(-Q)^{1/4}}\exp\!\big(\pm\tfrac{i}{\epsilon}\int^x\!\sqrt{-Q}\,\mathrm{d}x'\big)$	Oscillatory waves; in QM this is the propagating solution inside a well, $\|\psi\|^2\propto1/p$ .
Classically forbidden, $Q>0$	$y\sim \dfrac{C}{Q^{1/4}}\exp\!\big(\pm\tfrac{1}{\epsilon}\int^x\!\sqrt{Q}\,\mathrm{d}x'\big)$	Exponential growth/decay; the decaying branch is tunnelling penetration into a barrier.
Turning point, $Q(x_0)=0$	$\epsilon^2 y''\approx Q'(x_0)(x-x_0)\,y \Rightarrow$ Airy	WKB invalid; the local equation is Airy's, supplying the connection formulae.
Validity limit	$\big\|\epsilon Q'/Q^{3/2}\big\|\ll1 \Longleftrightarrow \big\|\mathrm{d}\lambda/\mathrm{d}x\big\|\ll1$	The wavelength must change little over a wavelength; fails within $\sim\epsilon^{2/3}$ of $Q=0$ .
Constant $Q$	$y=C_\pm e^{\pm\sqrt Q\,x/\epsilon}$ , WKB exact	With $Q'=0$ every correction $S_{n\ge2}$ vanishes; the exact plane wave is recovered.

Worked examples

1 · The harmonic oscillator, exactly (confidence builder)

Apply Bohr–Sommerfeld quantization to $V(x)=\tfrac12 m\omega^2 x^2$ and show it reproduces the exact spectrum.

Step 1. Locate the turning points. At energy $E$ the classical momentum is $p(x)=\sqrt{2m(E-\tfrac12 m\omega^2x^2)}$ , which vanishes at $x_\pm=\pm\sqrt{2E/(m\omega^2)}\equiv\pm a$ . The allowed region is $-a<x<a$ , with two simple turning points, so the Maslov-corrected rule with $+\tfrac12$ applies.

Step 2. Evaluate the action. $\displaystyle\int_{-a}^{a}\!\sqrt{2mE-m^2\omega^2x^2}\,\mathrm{d}x = m\omega\!\int_{-a}^{a}\!\sqrt{a^2-x^2}\,\mathrm{d}x = m\omega\cdot\tfrac{\pi a^2}{2}$ . Substituting $a^2=2E/(m\omega^2)$ gives the clean result $\displaystyle\int_a^b p\,\mathrm{d}x = m\omega\cdot\frac{\pi}{2}\cdot\frac{2E}{m\omega^2}=\frac{\pi E}{\omega}$ .

Step 3. Quantize. Bohr–Sommerfeld requires $\tfrac1\hbar\int_a^b p\,\mathrm{d}x=(n+\tfrac12)\pi$ , so $\dfrac{\pi E}{\hbar\omega}=(n+\tfrac12)\pi$ . Solving, $\boxed{E_n=(n+\tfrac12)\hbar\omega}$ . This is not an approximation for the oscillator — it is the exact quantum spectrum, and the $\tfrac12$ that everyone remembers as "zero-point energy" is here revealed as two $\pi/4$ turning-point phase shifts. A direct numerical quadrature of the action confirms $\int p\,\mathrm{d}x=(n+\tfrac12)\pi\hbar$ to five digits for $n=0,1,2,3$ .

2 · Tunnelling through a parabolic barrier (fully worked with numbers)

A particle of energy $E$ meets the inverted-parabola barrier $V(x)=V_0-\tfrac12 m\omega^2 x^2$ with $E<V_0$ . Estimate the transmission coefficient, then evaluate it for $V_0-E=2\hbar\omega$ .

Step 1. The WKB (Gamow) transmission coefficient is $T\approx e^{-2\gamma}$ with the barrier integral $\gamma=\tfrac1\hbar\int_{x_-}^{x_+}\!|p|\,\mathrm{d}x$ and $|p|=\sqrt{2m(V-E)}$ . The turning points are where $V=E$ , i.e. $x_\pm=\pm\sqrt{2(V_0-E)/(m\omega^2)}$ .

Step 2. The integral has exactly the same form as Example 1 with $E\to V_0-E$ : $\displaystyle\int_{x_-}^{x_+}\!\sqrt{2m(V_0-E)-m^2\omega^2x^2}\,\mathrm{d}x=\frac{\pi(V_0-E)}{\omega}$ . Therefore $2\gamma=\dfrac{2}{\hbar}\cdot\dfrac{\pi(V_0-E)}{\omega}=\dfrac{2\pi(V_0-E)}{\hbar\omega}$ , giving the compact result $\boxed{T\approx \exp\!\big(-2\pi(V_0-E)/\hbar\omega\big)}$ .

Step 3. Put in numbers. With $V_0-E=2\hbar\omega$ the exponent is $2\pi\cdot2=4\pi\approx12.566$ , so $T\approx e^{-4\pi}=\mathbf{3.5\times10^{-6}}$ . The barrier is only twice $\hbar\omega$ high yet suppresses transmission by six orders of magnitude — the hallmark exponential sensitivity of tunnelling. (The exact result for this smooth barrier is $T=1/(1+e^{4\pi})$ , equal to $3.5\times10^{-6}$ here; WKB captures the dominant exponential and errs only in the prefactor.)

Physics of the portrait

The WKB approximation is the rare method whose name lists three people while owing at least as much to three others — a nineteenth-century transformation rediscovered independently in 1926 by physicists chasing the new quantum mechanics.

Origins · 1 minWKB approximation and how this was discovered

WKB approximation

the semiclassical method y = exp(S/ε) for slowly varying linear ODEs · asymptotics and the classical limit of wave equations

The method carries the initials of Gregor Wentzel, Hendrik Kramers, and Léon Brillouin, who in 1926 independently applied it to Schrödinger's brand-new wave equation to extract the classical limit and recover Bohr–Sommerfeld quantization. But the mathematics is older and the credit contested. Joseph Liouville and George Green had, around 1837, developed the very transformation now used to solve slowly varying second-order equations — which is why mathematicians call the leading result the Liouville–Green approximation. Most pointedly, Harold Jeffreys published the connection formulae across turning points in 1923, three years before the physics trio, and in Britain the method is often called the "Jeffreys" or "WKBJ" approximation in his honour. The physicists supplied the decisive application — that $\hbar\to0$ is a singular limit and quantum mechanics reduces to classical mechanics exactly as wave optics reduces to rays — and George Gamow, with Gurney and Condon, immediately turned the forbidden-region branch into the first quantitative theory of alpha decay in 1928. The modern understanding, that the WKB series is a divergent asymptotic series whose two exponential branches trade dominance across Stokes lines, ties the whole subject back to Stokes's 1857 study of the Airy integral.

History

1837

— Liouville and Green develop the transformation and approximation for slowly varying second-order equations

1857

— Stokes studies the Airy integral and discovers the Stokes phenomenon at turning points

1923

— Jeffreys publishes the connection formulae linking oscillatory and exponential regions

1926

— Wentzel, Kramers, and Brillouin independently apply the method to the Schrödinger equation

1928

— Gamow, and independently Gurney and Condon, explain alpha decay by barrier tunnelling

Going further

Texts. Bender & Orszag, Advanced Mathematical Methods for Scientists and Engineers, Ch. 10 (the definitive treatment of WKB, turning points, and asymptotic matching) · Stone & Goldbart, Mathematics for Physics, chapter on WKB and asymptotics · Arfken, Weber & Harris, Mathematical Methods for Physicists, 7th ed. · Hassani, Mathematical Physics, on ODEs and asymptotic methods · Griffiths, Introduction to Quantum Mechanics, WKB chapter (the cleanest physics-first account) · Berry & Mount, Rep. Prog. Phys. 35 (1972), "Semiclassical approximations in wave mechanics" (the connection-formula and Stokes-phenomenon reference).

Provenance. The results are standard WKB/Liouville–Green theory. A verifier should reconfirm: (1) the Riccati reduction $(S')^2+\epsilon S''=Q$ and the transport result $S_1=-\tfrac14\ln Q$ (Bender & Orszag Ch. 10; Stone & Goldbart); (2) the Airy asymptotics $\mathrm{Ai}(z)\sim e^{-\frac23 z^{3/2}}/(2\sqrt\pi z^{1/4})$ and the $-\pi/4$ phase (Abramowitz & Stegun §10.4); (3) the harmonic-oscillator action $\int_{-a}^{a}p\,\mathrm{d}x=\pi E/\omega\Rightarrow E_n=(n+\tfrac12)\hbar\omega$ , checked numerically here to 5 digits for $n=0,\dots,3$ ; (4) the parabolic-barrier exponent $2\gamma=2\pi(V_0-E)/\hbar\omega$ , giving $T\approx e^{-4\pi}=3.5\times10^{-6}$ at $V_0-E=2\hbar\omega$ . No nonstandard numbers are used.

Builds on: Asymptotic Series and Poincaré Expansions · Order Symbols and Gauge Functions · Regular Perturbation Theory. Leads to: Connection Formulae and Turning Points · Bohr–Sommerfeld Quantization · Stokes Phenomenon and Anti-Stokes Lines · Optimal Truncation and Superasymptotics.

Ask the tutor: Ask me to walk through the connection formula at a real turning point, or to WKB-quantize any potential you name and compare the levels to the exact answer.

Is WKB the same thing as the semiclassical or geometric-optics approximation?

They are the same idea in different clothes. WKB is the general name for the $y=e^{S/\epsilon}$ expansion of a linear ODE with a slowly varying coefficient. When the ODE is Schrödinger's and the small parameter is $\hbar$ , it is the semiclassical approximation, and its eikonal equation is the classical Hamilton–Jacobi equation. When the ODE is Helmholtz's and the parameter is the wavelength, it is geometric optics and the eikonal is the ray equation. All three share the identical skeleton — eikonal phase at leading order, transport equation for the amplitude at next — so a result proved in one language transfers at once to the others.

Why is there a factor of two and a π/4 in the connection formula, exactly?

Both come from the exact asymptotics of the Airy function, and neither is adjustable. On the forbidden side the decaying tail of $\mathrm{Ai}(z)$ has amplitude $1/(2\sqrt\pi z^{1/4})$ ; on the allowed side its oscillatory tail has amplitude $1/(\sqrt\pi|z|^{1/4})$ — a factor of two larger — and it is a sine of $\tfrac23|z|^{3/2}+\pi/4$ , i.e. a cosine shifted by $\pi/4$ . Matching the WKB forms onto these two tails therefore forces exactly a factor of two in amplitude and a $-\pi/4$ in phase. These numbers are the local fingerprint of the Stokes phenomenon: they encode how the two exponential branches exchange dominance as you circle the turning point in the complex plane.

If the WKB series diverges, in what sense is it useful?

It is an asymptotic series, more useful here than a convergent one. For fixed $\epsilon$ the terms first shrink, reach a smallest term, then grow without bound; truncating at that smallest term gives the best approximation, with an error exponentially small in $1/\epsilon$ . So you cannot get arbitrary accuracy at fixed $\epsilon$ by adding terms, but you get superb accuracy for small $\epsilon$ by stopping at the right place. That irreducible error is not noise — it is the seed of the subdominant branch and the Stokes phenomenon, and the modern theory of resurgence shows it carries exact, recoverable information about the full solution.

When does WKB fail away from turning points?

Whenever the validity condition $|\epsilon Q'/Q^{3/2}|\ll1$ is violated, which can happen even where $Q\neq0$ . It fails near turning points (where $Q\to0$ makes the denominator blow up) but also wherever $Q$ has a sharp feature — a step, a cusp, a narrow well — that changes $Q$ appreciably within one wavelength. It also degrades near poles or higher-order zeros of $Q$ , where the simple linear-turning-point analysis gives way to another local special function (parabolic-cylinder functions for two nearby turning points, Bessel functions for confluent cases). Rule of thumb: WKB is trustworthy only where the potential is smooth on the wavelength scale.

How does WKB give tunnelling if the classical particle never enters the barrier?

Through the exponentially decaying branch in the forbidden region. Classically a particle with $E<V$ cannot exist where $Q>0$ , but the WKB solution there is not zero — it is $Q^{-1/4}e^{-\frac1\hbar\int|p|\,\mathrm{d}x}$ , an exponential leak of amplitude across the barrier. Connecting it to oscillatory solutions on both sides gives a small transmitted amplitude, and squaring it yields the Gamow factor $T\approx e^{-2\gamma}$ with $\gamma=\tfrac1\hbar\int|p|\,\mathrm{d}x$ . Because $\gamma$ sits in an exponent, tunnelling is extraordinarily sensitive to barrier height and width — the reason alpha-decay half-lives range over more than twenty orders of magnitude across the nuclear chart.

Why does the harmonic oscillator come out exact when WKB is only an approximation?

It is a happy accident of the quadratic potential, not a general feature. Bohr–Sommerfeld with the Maslov $+\tfrac12$ gives the exact spectrum $E_n=(n+\tfrac12)\hbar\omega$ , yet the WKB wavefunctions are still only approximate and diverge at the turning points. What is exact is the quantized energy, because for a quadratic potential the leading phase integral already contains all the information the exact quantization carries and the higher corrections happen to vanish. The Morse and Coulomb (with the Langer correction) problems are exact for related reasons, but a generic anharmonic potential carries a WKB energy error of order $\hbar^2$ that only higher terms remove.