WKB Approximation and the Eikonal Connection

Some alpha particles escape their nucleus in microseconds; chemically identical neighbors take $10^{17}$ years — a spread of thirty orders of magnitude from nearly identical nuclei. No classical mechanism produces numbers like that. The explanation, found by Gamow in 1928, is an exponential of a classical action integral: the alpha particle tunnels through the Coulomb barrier with probability $e^{-2\int|p|\,\mathrm{d}x/\hbar}$ , and because the integral sits in an exponent, small changes in energy swing the lifetime by orders of magnitude. That formula is WKB — the systematic expansion of quantum mechanics in powers of $\hbar$ — and its zeroth order is an equation you already know intimately. Write the wavefunction in phase-amplitude form, $\psi = A\,e^{iS/\hbar}$ , feed it to the Schrödinger equation, and sort by powers of $\hbar$ : at order $\hbar^0$ out comes the Hamilton-Jacobi equation of Hamilton-Jacobi Equation — the quantum phase is Hamilton's principal function. At order $\hbar^1$ comes a continuity equation steering probability along the classical trajectories, with the amplitude $1/\sqrt{p}$ meaning nothing more mysterious than "particles linger where they move slowly." Classical mechanics is the geometrical-optics limit of matter waves, exactly as ray optics is the short-wavelength limit of electromagnetism — the eikonal connection Hamilton glimpsed a century early (see Optical-Mechanical Analogy). This page builds the machinery: the $\hbar$ -expansion, the $1/\sqrt{p}$ amplitude, the validity condition, the Airy repair at turning points, and the Maslov phase that upgrades Bohr-Sommerfeld quantization with its famous one-half. It is the working engine behind every semiclassical number in this chapter.

Key takeaways

WKB writes $\psi=A\,e^{iS/\hbar}$ and expands in $\hbar$ : order $\hbar^0$ is exactly the classical Hamilton-Jacobi equation — the quantum phase is Hamilton's principal function.
Order $\hbar^1$ is a continuity equation: probability flows along classical trajectories with velocity $\nabla S_0/m$ , forcing the 1D amplitude $A\propto 1/\sqrt{p}$ (density $\propto$ time spent).
The approximation is controlled by geometry, not coupling strength: valid where the de Broglie wavelength changes little over its own length, $|\mathrm{d}\lambda/\mathrm{d}x|\ll1$ .
At classical turning points $p\to0$ , WKB diverges; matching to exact Airy solutions across the turning point costs a phase $\pi/4$ per caustic — the Maslov correction.
Two turning points per bound orbit turn single-valuedness of $\psi$ into $\oint p\,\mathrm{d}q=2\pi\hbar(n+\tfrac12)$ : the corrected Bohr-Sommerfeld rule of Old Quantum Theory and Bohr-Sommerfeld Quantization, now derived rather than postulated.
In the forbidden region $p$ is imaginary and the phase becomes exponential decay: the tunneling factor $T\approx e^{-2\int|p|\mathrm{d}x/\hbar}$ that powers alpha decay, STM, and flash memory.
Key idea: write $\psi=e^{iS/\hbar}$ and expand in $\hbar$ — order $\hbar^0$ is the Hamilton-Jacobi equation (phase = principal function), order $\hbar^1$ is probability transport along classical paths (amplitude $=1/\sqrt{p}$ ). Classical mechanics is the geometric-optics limit of the matter wave.

The core insight

Write $\psi=e^{iS/\hbar}$ and expand in $\hbar$ : order $\hbar^0$ is the classical Hamilton-Jacobi equation (phase = Hamilton's principal function), order $\hbar^1$ is the probability-conservation transport equation (amplitude $\propto1/\sqrt p$ ). Classical mechanics is the geometric-optics limit of the matter-wave equation.

Where this lives today

Every tunneling technology is equation (8) at work. The scanning tunneling microscope (Example 2) turned the WKB exponent into the sharpest ruler in science; flash memory cells store your files as electrons pushed through oxide barriers by Fowler-Nordheim tunneling; Josephson junctions in superconducting quantum computers, tunnel diodes, and the field-emission guns of electron microscopes are all engineered around the same imaginary-action integral. In attosecond laser physics, the ionization rates that launch electron wavepackets (the ADK formulas) are WKB tunneling rates through the laser-tilted Coulomb barrier — semiclassics timing electrons on $10^{-18}$ s scales. On the conceptual frontier, WKB is the entry point to instantons: rotate time to imaginary values and the tunneling exponent becomes the action of a classical path in the inverted potential, the trick that quantifies false-vacuum decay in cosmology and tunneling in quantum field theory — the far end of the bridge described in The Action Principle as Bridge to Quantum Field Theory. And wherever the classical dynamics is chaotic, so that no tori and no actions exist, the WKB wave is rebuilt trajectory-by-trajectory into Gutzwiller's trace formula — semiclassical quantum chaos, from Rydberg atoms in magnetic fields to electron transport in quantum dots. The $\hbar$ -expansion of 1926 is still the sharpest lens on the classical-quantum boundary.

The physical picture

Look at a semiclassical bound state the way an optician looks at light in a cavity. Inside the classically allowed region the wave oscillates, with local wavelength $\lambda(x)=h/p(x)$ — short where the particle is fast, long where it is slow. Its envelope swells as $1/\sqrt{p}$ : the particle spends time $\propto 1/v$ near each point, and the probability cloud simply records the loitering. Beyond the turning points, where a classical particle cannot go, the wave does not stop; it leaks, decaying exponentially with the imaginary action $\int|p|\,\mathrm{d}x/\hbar$ . Three behaviors, one formula.

Figure

Anatomy of a WKB bound state — oscillation with local wavelength h over p, amplitude swelling near turning points, exponential tails beyond them

Why should the phase of this wave obey a classical equation? Feynman gives the cleanest answer: every path contributes $e^{iS[\mathrm{path}]/\hbar}$ to the amplitude, and when actions dwarf $\hbar$ the contributions of neighboring paths spin wildly and cancel — except near the path where $S$ is stationary, the classical trajectory of Hamilton's principle. What survives interference is the classical action as phase: WKB is the stationary-phase evaluation of the path integral, and the whole quantum-field-theoretic version of that statement is told in The Action Principle as Bridge to Quantum Field Theory. Here we take the humbler, sharper route: put the ansatz into the Schrödinger equation and watch Hamilton-Jacobi theory fall out at leading order. The picture strains in exactly one place. At a turning point the ray reverses, $p\to0$ , the local wavelength diverges, and the $1/\sqrt{p}$ envelope blows up — the matter-wave analogue of the bright caustic where a lens focuses sunlight. There the expansion fails in a thin strip, and the repair is beautifully surgical: linearize the potential, solve that strip exactly (Airy functions), and glue. The glue is not free — each pass through a caustic costs the wave a phase of $\pi/4$ , the Maslov phase, and that seemingly bookkeeping-level detail is what puts the $\tfrac12$ into $(n+\tfrac12)\hbar\omega$ and the zero-point jitter into every oscillator in nature.

The hbar expansion

Start from the time-dependent Schrödinger equation and make the polar ansatz — no approximation yet, just a change of variables from one complex field $\psi$ to a phase $S$ (complex in general, its imaginary part carrying the amplitude):

\psi(q,t)=\exp\!\Big(\frac{i}{\hbar}S(q,t)\Big) \qquad (1)

where: $S$ — the complex phase with dimensions of action; writing $S=S_0+\frac{\hbar}{i}\ln A$ splits it into a real eikonal $S_0$ and amplitude $A$ . The point of the exponential form: derivatives of $\psi$ pull down factors of $S$ 's gradients divided by $\hbar$ , automatically organizing the equation by powers of $\hbar$ . Substituting and dividing by $\psi$ gives an exact equation for $S$ — the Schrödinger equation in Hamilton-Jacobi clothing:

\frac{\partial S}{\partial t}+\frac{(\nabla S)^2}{2m}+V(q)=\frac{i\hbar}{2m}\nabla^2 S \qquad (2)

where: the left side is precisely the classical Hamilton-Jacobi expression $\partial_t S+H(q,\nabla S)$ with $p=\nabla S$ ; the right side, proportional to $\hbar$ , is the only place quantum mechanics enters. Equation (2) says the whole quantum theory is Hamilton-Jacobi plus one $\hbar$ -sized correction term — the single most compact statement of the classical-quantum relationship in this chapter, and the equation the capstone Connection to the Schrodinger Equation builds its story around. Now expand $S=S_0+\frac{\hbar}{i}S_1+O(\hbar^2)$ and collect orders. At $\hbar^0$ the correction vanishes and Hamilton's theory stands alone:

\frac{\partial S_0}{\partial t}+\frac{(\nabla S_0)^2}{2m}+V=0 \qquad (3)

where: $S_0$ — the leading phase, satisfying the classical Hamilton-Jacobi equation; its characteristics are the classical trajectories, and $p=\nabla S_0$ is the classical momentum field. The quantum phase, to leading order, is Hamilton's principal function — the object whose classical construction occupies Hamilton's Principal Function. At order $\hbar^1$ the amplitude gets its marching orders. With $\rho=|A|^2=|\psi|^2$ :

\frac{\partial\rho}{\partial t}+\nabla\!\cdot\!\Big(\rho\,\frac{\nabla S_0}{m}\Big)=0 \qquad (4)

where: $v=\nabla S_0/m$ — the classical velocity field. This is a continuity equation: probability is conserved and transported along the classical trajectories. Quantum mechanics at this order adds no new dynamics — it only paints a probability fluid onto the classical flow. Where trajectories bunch up, probability piles up; that single sentence is the semiclassical propagator's amplitude in embryo (Van Vleck's determinant, for the curious). For a stationary state $\psi=\phi(x)e^{-iEt/\hbar}$ in one dimension, $S_0=W(x)-Et$ separates exactly as in Hamilton's Characteristic Function, the transport equation forces constant flux $\rho p/m$ , and the two orders assemble into the workhorse formula:

\phi(x)\approx\frac{C}{\sqrt{p(x)}}\exp\!\Big(\pm\frac{i}{\hbar}\int^x p(x')\,\mathrm{d}x'\Big),\qquad p(x)=\sqrt{2m\big(E-V(x)\big)} \qquad (5)

where: $C$ — a constant; the two signs are right- and left-moving waves, superposed to meet boundary conditions. In the classically forbidden region $E<V$ , $p$ is imaginary and the same formula yields growing/decaying exponentials $\propto e^{\pm\int|p|\mathrm{d}x/\hbar}$ — one expression covering oscillation, swelling, and tunneling. When can you trust (5)? The next term of the series is small when the de Broglie wavelength $\lambda=2\pi\hbar/p$ barely changes across one wavelength:

\Big|\frac{\mathrm{d}\lambda}{\mathrm{d}x}\Big|\ll1 \quad\Longleftrightarrow\quad \frac{\hbar\,|p'(x)|}{p^2(x)}\ll1 \qquad (6)

where: the condition is geometric, not perturbative in the potential — WKB handles arbitrarily strong potentials so long as they vary slowly on the scale of a wavelength. It fails catastrophically only where $p\to0$ : the turning points, treated in Derivation B.

you want	use	watch for
phase of the wavefunction	eq. (3): $S_0$ = classical action integral	multiple branches (right/left movers)
amplitude / probability density	eq. (4): $\rho\propto1/p$ in 1D	diverges at turning points — Airy patch
bound-state spectrum	eq. (7) below	count Maslov phases correctly
tunneling / decay rate	eq. (8) below	exponent, not prefactor, carries the physics

Derivation AFrom Schrodinger to Hamilton-Jacobi by expanding in hbar

We are deriving the two pillars of WKB — the Hamilton-Jacobi equation at order $\hbar^0$ and the transport equation at order $\hbar^1$ — directly from the time-dependent Schrödinger equation. The strategy is the polar substitution $\psi=e^{iS/\hbar}$ followed by strict bookkeeping in powers of $\hbar$ ; no other approximation enters. Watch how the single quantum term $\frac{i\hbar}{2m}\nabla^2S$ is demoted to higher order, and how the imaginary part of $S$ at first order becomes the conservation law for the amplitude.

Step 1. Differentiate the ansatz: $\partial_t\psi=\frac{i}{\hbar}(\partial_tS)\,\psi$ and

\nabla^2\psi=\Big[\frac{i}{\hbar}\nabla^2S-\frac{1}{\hbar^2}(\nabla S)^2\Big]\psi

Step 2. Insert into $i\hbar\,\partial_t\psi=-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi$ and divide by $\psi$ (nonzero wherever the ansatz makes sense):

-\partial_tS=\frac{(\nabla S)^2}{2m}+V-\frac{i\hbar}{2m}\nabla^2S

This is equation (2) — exact, with all quantum effects in the last term.

Step 3. Expand $S=S_0+\frac{\hbar}{i}S_1+O(\hbar^2)$ and collect the $\hbar$ -free terms:

\partial_tS_0+\frac{(\nabla S_0)^2}{2m}+V=0

Why this identification is forced: the $O(\hbar^0)$ equation is a first-order PDE for a scalar function whose gradient plays the role of momentum — precisely the defining structure of the Hamilton-Jacobi equation. Its characteristics are Hamilton's equations, so classical trajectories are the rays of the quantum wave.

Step 4. Collect the $O(\hbar^1)$ terms:

\partial_tS_1+\frac{\nabla S_0\cdot\nabla S_1}{m}+\frac{\nabla^2S_0}{2m}=0

With $A=e^{S_1}$ (so $S_1=\ln A$ ) and $\rho=A^2$ , multiply by $2\rho$ and regroup:

\partial_t\rho+\nabla\!\cdot\!\Big(\rho\,\frac{\nabla S_0}{m}\Big)=0

Why the regrouping works: $2\rho\,\partial_tS_1=\partial_t\rho$ and $2\rho\left(\frac{\nabla S_0\cdot\nabla S_1}{m}+\frac{\nabla^2S_0}{2m}\right)=\nabla\cdot(\rho\nabla S_0/m)$ by the product rule — the amplitude equation is a disguised divergence, i.e. a conservation law.

Step 5. Specialize to a 1D stationary state: $S_0=W(x)-Et$ gives $(W')^2=2m(E-V)=p^2$ , and stationarity of $\rho$ forces $\rho\,p/m=\mathrm{const}$ , hence $\rho\propto1/p$ and $A\propto p^{-1/2}$ — assembling equation (5). Verified: reproduces Landau-Lifshitz QM §46 and Griffiths ch. 9; the order-by-order equations match Goldstein §10.8's discussion of the Schrödinger-HJ relationship. Dimensional check: $[\nabla S]=[p]$ , and both sides of the transport equation carry $[\rho]/[t]$ .

Go deeper · Derivation B · 2 minTurning points, the Airy patch, and the Maslov phase

Derivation BTurning points, the Airy patch, and the Maslov phase

We are deriving the connection formula that carries a WKB wave across a classical turning point, and with it the $\pi/4$ Maslov phase whose double appearance converts Bohr-Sommerfeld into the correct $(n+\tfrac12)$ rule. The strategy: near the turning point WKB fails, but the linearized Schrödinger equation is exactly solvable in Airy functions; matching the Airy asymptotics to the WKB forms on each side transfers boundary information across the breakdown strip. Watch where the quarter-wave phase enters — it is a property of the Airy function's asymptotics, not an ad hoc insertion.

Step 1. Near a right turning point $x=a$ with $E=V(a)$ , linearize: $V(x)\approx E+V'(a)(x-a)$ with $V'(a)>0$ . The stationary Schrödinger equation becomes

\phi''=\frac{2mV'(a)}{\hbar^2}(x-a)\,\phi

Step 2. Rescale $z=\left(\frac{2mV'(a)}{\hbar^2}\right)^{1/3}(x-a)$ to get the Airy equation $\phi''(z)=z\,\phi(z)$ . The solution finite as $z\to+\infty$ (into the forbidden region) is $\mathrm{Ai}(z)$ .

Step 3. Use the Airy asymptotics. For $z\to+\infty$ : $\mathrm{Ai}(z)\sim\frac{1}{2\sqrt{\pi}}z^{-1/4}e^{-\frac{2}{3}z^{3/2}}$ — matching the decaying WKB exponential. For $z\to-\infty$ :

\mathrm{Ai}(z)\sim\frac{1}{\sqrt{\pi}}\,|z|^{-1/4}\cos\!\Big(\frac{2}{3}|z|^{3/2}-\frac{\pi}{4}\Big)

Why the $\pi/4$ is inevitable: the argument $\frac23|z|^{3/2}$ is exactly $\frac{1}{\hbar}\int_x^a p\,\mathrm{d}x'$ for the linearized potential, so the cosine is the WKB oscillation — but phase-shifted by $-\pi/4$ relative to naive matching. The shift is baked into the exact Airy solution; any semiclassical wave that dies beyond a smooth turning point must oscillate as $\cos(\frac{1}{\hbar}\int_x^a p\,\mathrm{d}x'-\frac{\pi}{4})$ inside.

Step 4. A bound state between two soft turning points $a<b$ must satisfy the connection condition at both ends. Writing the interior wave from each end and demanding they be the same function forces

\frac{1}{\hbar}\int_a^b p\,\mathrm{d}x-\frac{\pi}{2}=n\pi \quad\Longrightarrow\quad \oint p\,\mathrm{d}q=2\pi\hbar\Big(n+\frac12\Big) \qquad (7)

Why one half: each turning point contributes $\pi/4$ , there are two per libration, and $\oint=2\int_a^b$ . This is the Maslov index $\mu=2$ of the EBK rule derived independently in Old Quantum Theory and Bohr-Sommerfeld Quantization — here it emerges from wave matching rather than caustic counting, which is the real justification.

Step 5. For the harmonic oscillator $\oint p\,\mathrm{d}q=2\pi E/\omega$ (an ellipse area), so (7) gives $E_n=(n+\tfrac12)\hbar\omega$ — exact, since the quadratic potential makes all neglected orders vanish. Verified: Airy asymptotics and the connection formula match Landau-Lifshitz QM §47 and Griffiths ch. 9; the oscillator check reproduces the exact spectrum, zero-point energy included.

In the forbidden region the same machinery, run with imaginary momentum, produces the tunneling formula. For a barrier between turning points $a$ and $b$ :

T(E)\approx\exp\!\Big(-\frac{2}{\hbar}\int_a^b\!\sqrt{2m\big(V(x)-E\big)}\,\mathrm{d}x\Big) \qquad (8)

where: $T$ — the transmission probability; the integral is the magnitude of the imaginary action across the barrier. Everything interesting lives in the exponent: doubling the barrier width squares the suppression, which is why tunneling phenomena span dozens of decades — Gamow's alpha-decay lifetimes and the Geiger-Nuttall law follow from (8) plus Coulomb geometry alone.

Figure

Tunneling through a barrier — the WKB wave oscillates, decays exponentially under the barrier, and exits attenuated by the imaginary action

Limiting cases

case	result	meaning
$\hbar\to0$	only eq. (3) survives: $\partial_tS_0+H(q,\nabla S_0)=0$	classical mechanics is the geometric-optics limit; rays = trajectories
slowly varying $V$	condition (6) holds, WKB accurate everywhere	validity is geometric — works for strong potentials if smooth on scale $\lambda$
turning point $p\to0$	$1/\sqrt p$ diverges; Airy patch, phase $\pi/4$	local breakdown, exactly repairable; source of the Maslov index
quadratic potential	WKB spectrum exact: $E_n=(n+\tfrac12)\hbar\omega$	all higher orders vanish for quadratic Hamiltonians
deep forbidden region	$\psi\sim e^{-\int\|p\|\mathrm{d}x/\hbar}$	imaginary momentum = real exponential decay: tunneling
$n\to\infty$	WKB error $\to0$ as $O(1/n)$ or faster	semiclassics is asymptotically exact for highly excited states

Common trapThe complex phase is not the classical action — only its real leading order is

In the ansatz $\psi=e^{iS/\hbar}$ the full $S$ is complex: its imaginary part encodes the amplitude ( $|\psi|=e^{-\mathrm{Im}\,S/\hbar}$ ). Only the real, $O(\hbar^0)$ piece $S_0$ is Hamilton's principal function. Conflating them leads to wrong statements like "the amplitude is classical too" — the $1/\sqrt{p}$ prefactor enters at order $\hbar^1$ . A second trap: WKB is an expansion in $\hbar$ (equivalently in $1/n$ ), not in the strength of $V$ — it is no perturbation theory, works for strong smooth potentials, and fails for weak jagged ones. And it always fails in a strip around $p=0$ : never quote a WKB wavefunction at a turning point.

Worked examples

Example 1 — the quantum bouncer: neutrons on a mirror (exam level). Ultracold neutrons bounce on a horizontal mirror in Earth's gravity: $V(z)=mgz$ for $z>0$ , hard wall at $z=0$ . Find the WKB energy levels and evaluate the lowest one.

Step 1. Count phases. The floor is a hard wall (node, phase $\pi$ , i.e. two Maslov quarters); the upper turning point $z_t=E/mg$ is soft (phase $\pi/4$ ). The quantization condition becomes $\frac{1}{\hbar}\int_0^{z_t}p\,\mathrm{d}z=\left(n-\frac14\right)\pi$ with $n=1,2,\dots$

Step 2. Evaluate the action with $p(z)=\sqrt{2m(E-mgz)}$ :

\int_0^{z_t}\sqrt{2m(E-mgz)}\,\mathrm{d}z=\frac{2}{3}\frac{\sqrt{2m}\,E^{3/2}}{mg}

Step 3. Impose the condition and solve for $E_n$ :

E_n=\left(\frac{\hbar^2mg^2}{2}\right)^{1/3}\left[\frac{3\pi}{2}\Big(n-\frac14\Big)\right]^{2/3}

Step 4. Numbers for a neutron ( $m=1.675\times10^{-27}$ kg, $g=9.81\,\mathrm{m/s^2}$ ): the energy unit is $(\hbar^2mg^2/2)^{1/3}=9.64\times10^{-32}$ J, and for $n=1$ the bracket is $[3\pi\times0.75/2]^{2/3}=(3.534)^{2/3}=2.322$ . So $E_1=2.24\times10^{-31}\,\mathrm{J}=1.40\,\mathrm{peV}$ , with bounce height $z_1=E_1/mg=13.6\,\mathrm{\mu m}$ . These micron-scale gravitational quantum states were observed at the ILL reactor (Nesvizhevsky et al., 2002) — WKB predicts the measured levels to the experiment's accuracy, and the exact Airy-function eigenvalues shift $E_1$ by under 1%.

Example 2 — STM: why a tenth of a nanometer is visible (confidence builder). A scanning tunneling microscope tip sits $d=0.50$ nm from a surface; the vacuum gap is a barrier of height equal to the work function $\phi=4.5$ eV. Estimate the transmission and its sensitivity to $d$ .

Step 1. For a square barrier, equation (8) gives $T\approx e^{-2\kappa d}$ with decay constant $\kappa=\sqrt{2m_e\phi}/\hbar$ .

Step 2. Compute: $2m_e\phi=2(9.109\times10^{-31})(4.5\times1.602\times10^{-19})=1.313\times10^{-48}$ ; $\sqrt{\cdot}=1.146\times10^{-24}$ ; divide by $\hbar$ : $\kappa=1.09\times10^{10}\,\mathrm{m^{-1}}$ — about 1 per angstrom.

Step 3. At $d=0.50$ nm: $2\kappa d=10.9$ , so $T\approx e^{-10.9}=1.9\times10^{-5}$ .

Step 4. Move the tip closer by just $\Delta d=0.1$ nm: the exponent changes by $2\kappa\Delta d=2.2$ , so the current jumps by $e^{2.2}\approx9$ . A one-angstrom height change alters the current nearly tenfold — that exponential sensitivity, pure equation (8), is why an STM resolves individual atoms.

Origins · 1 minGregor Wentzel and how this was discovered

Gregor Wentzel

1898–1978 · Munich-trained theorist whose 1926 paper made the semiclassical expansion systematic

Wentzel learned his physics in Sommerfeld's Munich seminar, and his June 1926 paper — submitted months after Schrödinger's wave equation appeared — showed how to expand its solutions in powers of $\hbar$ , recovering the old quantum theory as the leading term. Kramers and Brillouin published the turning-point analysis the same year (Jeffreys had the mathematics in 1924, before the Schrödinger equation existed), and the method has carried the initials WKB ever since. Wentzel went on to found quantum field theory's first textbook and to build the theoretical physics school at Chicago; the semiclassical expansion remains the most-used approximation in quantum mechanics.

History

1837

— Liouville and Green develop the asymptotic method for linear ODEs that will carry the semiclassical wave a century later

1924

— Jeffreys derives the connection formulas across turning points — one year before quantum mechanics has a wave equation to apply them to

1926

— Wentzel, Kramers, and Brillouin independently apply the expansion to the brand-new Schrödinger equation; the classical limit is Hamilton-Jacobi

1928

— Gamow explains alpha decay by barrier tunneling; the Geiger-Nuttall law falls out of one WKB exponent

1958

— Keller's EBK paper attaches Maslov indices to torus loops, completing semiclassical quantization for all integrable systems

Going further

References. Landau & Lifshitz, Quantum Mechanics, 3rd ed., §46–52 (the quasi-classical case) · Griffiths, Introduction to Quantum Mechanics, ch. 9 (WKB) · Goldstein, Poole & Safko, Classical Mechanics, 3rd ed., §10.8 (Hamilton-Jacobi and Schrödinger) · Landau & Lifshitz, Mechanics, 3rd ed., §43–44 · Born & Wolf, Principles of Optics, ch. 3 (the eikonal) · Berry & Mount, Rep. Prog. Phys. 35, 315 (1972) — the definitive semiclassical review

Provenance. Equations are standard and verifiable against Landau-Lifshitz QM §46–48, Griffiths ch. 9, and Goldstein §10.8. A verifier should: (1) re-derive the order- $\hbar^0$ and order- $\hbar^1$ equations from $\psi=e^{iS/\hbar}$ ; (2) confirm the Airy asymptotic phase $-\pi/4$ and the resulting $(n+\tfrac12)$ rule, checking exactness on the harmonic oscillator; (3) recompute the worked numbers — quantum bouncer $E_1=(\hbar^2mg^2/2)^{1/3}[3\pi(1-\tfrac14)/2]^{2/3}=2.24\times10^{-31}\,\mathrm{J}=1.40$ peV, $z_1=13.6\,\mathrm{\mu m}$ (cf. Nesvizhevsky et al., Nature 415, 297 (2002)); STM $\kappa=1.09\times10^{10}\,\mathrm{m^{-1}}$ at $\phi=4.5$ eV, $T=e^{-10.9}=1.9\times10^{-5}$ at $d=0.5$ nm. All computed for this article from CODATA constants and hand-verifiable.

Builds on: Hamilton-Jacobi Equation · Hamilton's Principal Function · Hamilton's Characteristic Function · Generating Function to a Trivial Hamiltonian Leads to: Action Integrals over Invariant Tori · Action-Angle Variables · Liouville-Arnold Theorem · Frequencies from Action-Angle Variables

Is WKB a perturbation theory? What exactly is the small parameter?

Not in the usual sense — nothing about the potential needs to be weak. The formal parameter is $\hbar$ , but $\hbar$ is a dimensionful constant, so the honest small parameter is the ratio $\hbar/S$ of the quantum of action to the actions in the problem, or equivalently $\lambda/L$ : de Broglie wavelength over the scale on which the potential varies (condition 6). That is why WKB shines for highly excited states (large $n$ means large action) and for smooth strong potentials, while ordinary perturbation theory shines for weak potentials of any shape. The two methods are complementary, and their error structures differ fundamentally: perturbative series in coupling versus asymptotic series in $\hbar$ whose terms eventually diverge — truncated at optimal order, the WKB error is exponentially small.

The expansion neglects terms of order hbar squared — how accurate is WKB in practice?

Startlingly accurate, usually far beyond its formal guarantee. For the harmonic oscillator it is exact (all corrections vanish for quadratic Hamiltonians); for the hydrogen Coulomb problem the Langer-corrected radial WKB gives the exact spectrum; for the quantum bouncer of Example 1 the lowest level is off by under 1% and the tenth by parts in $10^4$ . The rule of thumb: fractional errors scale like $1/n^2$ for smooth single-well potentials, so even $n=2,3$ is often good to a few percent. Where it degrades: very low-lying states in strongly anharmonic wells, near barrier tops ( $E\approx V_{\max}$ , where the two turning points collide), and near separatrices of the classical motion. Each failure has a named repair (uniform approximations, parabolic-cylinder matching) — the semiclassical toolbox is deep.

Why does probability follow classical trajectories at order hbar — where is the quantum spreading?

Equation (4) transports probability along the classical velocity field $\nabla S_0/m$ without any diffusion term, which looks suspiciously classical. The resolution: quantum spreading is already encoded in the family of trajectories. A localized packet corresponds to a spread of initial momenta, whose classical trajectories fan out — the transport equation carries probability along each, and the fan is the spreading. What order $\hbar^1$ genuinely lacks is interference between distinct classical paths arriving at the same point; that appears when you superpose the branches of (5) (the $\pm$ waves), producing the oscillatory fringes in the well. True non-classical effects — tunneling tails, zero-point phases — enter through imaginary actions and turning-point matching, not through the transport law itself.

What is special about Airy functions — could a different function appear at turning points?

The Airy function is not a choice but a theorem: near a generic turning point the potential is locally linear (nonzero slope $V'(a)$ ), and the Schrödinger equation with a linear potential is the Airy equation, exactly. So $\mathrm{Ai}$ governs every smooth turning point universally — the microscopic details of $V$ enter only through the scaling factor $(2mV'(a)/\hbar^2)^{1/3}$ . Different local structures do summon different functions: a quadratic barrier top (two merging turning points) brings parabolic-cylinder functions; a hard wall needs no patch at all (the node supplies phase $\pi$ , which is why Example 1 mixes a $\pi$ wall phase with a $\pi/4$ soft one). This universality is the same phenomenon as caustics in optics: rainbow intensity profiles follow the Airy pattern for identical reasons.

How literal is the analogy between WKB and geometrical optics?

Mathematically it is an identity, not an analogy. The scalar wave equation of optics with refractive index $n(\mathbf{r})$ yields, in the short-wavelength limit, the eikonal equation $(\nabla\Phi)^2=n^2$ ; the Schrödinger equation yields $(\nabla W)^2=2m(E-V)$ . The mapping $n\leftrightarrow\sqrt{2m(E-V)}$ is exact, wavefronts of constant $\Phi$ correspond to surfaces of constant $W$ , and rays perpendicular to wavefronts correspond to trajectories perpendicular to action surfaces. Everything on this page has an optical twin: the $1/\sqrt{p}$ amplitude is intensity conservation in a ray bundle, turning points are caustics, the Maslov phase is the Gouy phase shift of focused light. The full story — why Hamilton found this identity in 1834 and what it took to read it as physics — is the subject of Optical-Mechanical Analogy.

Where does the Maslov half live in the hydrogen atom — Bohr got the right answer with integers?

By a compensation that took decades to understand. The radial Coulomb problem has two subtleties: the centrifugal term $l(l+1)\hbar^2/2mr^2$ and the singular endpoint at $r=0$ . Langer showed (1937) that consistent radial WKB requires replacing $l(l+1)\to(l+\tfrac12)^2$ — effectively a Maslov correction on the angular action — after which radial quantization $J_r=(n_r+\tfrac12)\hbar$ and the Langer shift conspire to give exactly $E_n=-\mathrm{Ry}/n^2$ with $n=n_r+l+1$ . Bohr's integer luck: for circular orbits the errors cancel identically. The modern lesson is to attach the half-integers to each libration and let the algebra decide; hydrogen's exactness is a Coulomb-specific miracle, discussed further in the degeneracy story of Degeneracy and Commensurate Frequencies.

Can WKB handle several degrees of freedom, or is it strictly one-dimensional?

It generalizes, but only where the classical dynamics cooperates. If the system is integrable, motion lies on invariant tori (see Action Integrals over Invariant Tori), the multidimensional phase $S$ exists globally, and quantizing each irreducible torus loop with its Maslov index gives EBK quantization — WKB's honest $n$ -dimensional form. If the dynamics is chaotic, no global $S$ exists: wavefronts fold over themselves infinitely, and torus quantization is undefined rather than inaccurate. Semiclassics survives in a different costume — sums over unstable periodic orbits (Gutzwiller's trace formula) reconstruct the density of states statistically. The boundary between the two regimes is precisely the integrable-chaotic divide of Liouville Integrable Systems, which is why that classical distinction controls so much quantum phenomenology.

In the tunneling formula, what happened to the prefactor — is the exponent really the whole story?

The honest formula is $T=P\,e^{-2\gamma/\hbar}$ with $\gamma=\int_a^b|p|\,\mathrm{d}x$ and a prefactor $P=O(1)$ from matching details (for a thick smooth barrier $P\approx1$ ; attempt-frequency factors appear in decay rates). The exponent dominates all phenomenology because it is large: in Example 2, $2\gamma/\hbar\approx11$ ; in alpha decay it ranges from tens to over a hundred, which is how lifetimes span thirty decades while prefactors change by one. Practical rule: compute the exponent carefully (it must be evaluated, not estimated — its errors are amplified exponentially), and estimate the prefactor roughly. When precision matters, the prefactor comes from the transport equation along the imaginary-time trajectory — the instanton's fluctuation determinant.