The WKB Approximation

The WKB approximation is quantum mechanics in the short-wavelength limit — the wave analogue of geometrical optics. Just as light rays emerge from Maxwell's equations when the wavelength is tiny compared with the scale over which the medium changes, classical trajectories emerge from the Schrödinger equation when the de Broglie wavelength is short compared with the scale over which the potential varies. It is the workhorse for approximate energy levels and tunneling rates in one-dimensional (or separable) problems where the potential is smooth but no exact solution exists. The construction is a single, disciplined move. Write the stationary wavefunction as $\psi(x)=\exp[iS(x)/\hbar]$ and substitute into the time-independent Schrödinger equation. This turns a linear equation into a nonlinear one for $S$ , which we then expand in powers of $\hbar$ . The leading term recovers the classical action $S_0=\int p\,\mathrm{d}x$ ; the next term fixes the amplitude to $p^{-1/2}$ , exactly the factor that conserves probability flux. Both orders are derived here from scratch. The result is a locally-plane wave whose crests track the classical momentum: $\psi\approx p^{-1/2}\exp[\pm(i/\hbar)\int p\,\mathrm{d}x]$ . That wavefunction is the deliverable this article owns. We also own its precise domain of validity — the slowly-varying condition $|\mathrm{d}\lambda/\mathrm{d}x|\ll 1$ , derived rather than asserted — and the exact place it fails: the classical turning points, where $p\to 0$ and the $p^{-1/2}$ amplitude diverges. We show the divergence explicitly and name the cure without stealing its derivation: the Airy-function patch and the connection formulae belong to a companion article. What comes next is deferred by name, not repeated. The quantization condition $\oint p\,\mathrm{d}x=(n+\tfrac12)h$ is set up here and derived next door; the connection formulae that carry the $\pi/4$ phases are their own article; the tunneling Gamow factor is a third. This is the on-ramp — the wavefunction, its amplitude, its wavelength, its breakdown — that all three consume.

Key takeaways

Write $\psi=e^{iS/\hbar}$ and expand $S=S_0+\hbar S_1+\cdots$ : the leading order gives the classical action $S_0=\int p\,\mathrm{d}x$ , the next order gives the flux-conserving amplitude $p^{-1/2}$ .
In classically allowed regions ( $E>V$ ) the wave oscillates as $p^{-1/2}\exp(\pm\tfrac{i}{\hbar}\int p\,\mathrm{d}x)$ ; in forbidden regions ( $E<V$ ) it grows or decays as $|p|^{-1/2}\exp(\pm\tfrac{1}{\hbar}\int|p|\,\mathrm{d}x)$ , with $p=\sqrt{2m(E-V)}$ .
The amplitude $\propto p^{-1/2}$ means $|\psi|^2\propto 1/p$ — the particle is most likely found where it moves slowest, the semiclassical bridge to classical dwell-time probability.
Validity requires the local de Broglie wavelength to change slowly, $|\mathrm{d}\lambda/\mathrm{d}x|\ll 1$ ; this ALWAYS fails at a turning point where $p\to0$ , no matter how smooth the potential.
The Airy-function patch and the connection formulae that repair the turning point are their own article; the quantization condition and the Gamow factor are set up here and derived in companion articles.
WKB is the leading term of a small- $\hbar$ (large-action) expansion and joins quantum mechanics smoothly onto classical Hamilton–Jacobi theory.
Key idea: $\psi=e^{iS/\hbar}$ with $S$ expanded in $\hbar$ turns the Schrödinger equation into a classical action plus a $p^{-1/2}$ amplitude — a locally-plane matter wave, exact in the bulk and broken only at turning points.

The core insight

Write $\psi=e^{iS/\hbar}$ and expand $S$ in powers of $\hbar$ . The leading order is the classical action $\int p\,\mathrm{d}x$ ; the next order fixes the amplitude to $p^{-1/2}$ , conserving flux. The result is a locally-plane wave whose wavelength tracks the classical momentum — excellent in the bulk of a well or barrier, and divergent only in small neighbourhoods of the turning points.

Where this lives today

The WKB wavefunction is the entry point to modern semiclassics. Its immediate children — quantization, connection formulae, tunneling — are the companion articles in this chapter. Beyond them, the same locally-plane-wave idea generalizes to the EBK (Einstein–Brillouin–Keller) torus quantization of integrable systems, to Gutzwiller's trace formula for chaotic ones, and to the stationary-phase evaluation of path integrals where the classical action again sits in the exponent. In applied settings the amplitude-tracks-momentum picture underlies ray tracing in optics and acoustics, adiabatic invariants in plasma physics, and the semiclassical propagation methods used in chemical dynamics. The grad-register treatment — higher-order $\hbar$ corrections, uniform approximations, and multidimensional tori — is the subject of The WKB Approximation at graduate level, which recalls this article by name and extends it; the deep validity-and-breakdown analysis is deferred to Validity and Breakdown of WKB.

The physical picture

Think of WKB as quantum geometrical optics. A wavefunction $e^{iS/\hbar}$ has a local wavenumber $k(x)=S'/\hbar=p(x)/\hbar$ , so its crests march in step with the classical momentum. Where the particle is fast (large $p$ ) the wavelength is short and the crests bunch up; where it is slow (near a turning point) the wavelength stretches. The amplitude does the opposite of the speed: the $p^{-1/2}$ factor makes $|\psi|^2\propto 1/p$ , piling probability up exactly where a classical particle would linger. This is the semiclassical dictionary — the quantum probability density becomes the classical fraction of time spent. The first figure makes the whole story visible in one frame. The oscillation is real WKB, its local wavelength widening toward the edges of the well; the dashed envelope is $\pm p^{-1/2}$ . Follow the envelope to the turning points and watch it climb — it is heading for infinity. That divergence is not a numerical artefact and not a defect of a particular potential: it is built into $p^{-1/2}$ the instant $p\to0$ , and it is the single reason WKB needs surgical repair at the edges.

Figure

The WKB wave and its 1/√p envelope: perfect in the bulk, diverging at both turning points

The second reading is the validity condition. WKB demands that the wavelength change by much less than itself over one wavelength, $|\mathrm{d}\lambda/\mathrm{d}x|\ll1$ . In the middle of a deep well this is superbly satisfied; near a turning point it fails catastrophically because $\lambda=2\pi\hbar/p\to\infty$ . So the method is not "approximately right everywhere" — it is essentially exact in the bulk and simply inapplicable in tiny turning-point neighbourhoods, which is why the fix is local surgery rather than a global correction. The optics analogy is worth drawing out, because it is where the name and the intuition come from. A short-wavelength light wave in a medium of slowly-varying refractive index reduces to rays that bend according to Fermat's principle; a short-de-Broglie-wavelength matter wave in a slowly-varying potential reduces to classical trajectories obeying Hamilton's principle. In both, the leading phase is a path-length-like quantity (the optical path, or the classical action), and the amplitude adjusts to conserve flux. WKB is literally the eikonal approximation of quantum mechanics.

Figure

Quantum geometrical optics: wavelength tracks momentum, and ħ→0 hands back the classical ray

The core equations

We start from the stationary Schrödinger equation and extract the WKB wavefunction by an $\hbar$ -expansion of the phase. The single approximation is that the series in $\hbar$ truncates well — everything else is exact algebra. The time-independent Schrödinger equation for a particle of mass $m$ in a potential $V(x)$ is the object we approximate.

-\frac{\hbar^2}{2m}\psi''(x)+V(x)\,\psi(x)=E\,\psi(x) \qquad (1)

where: $\psi(x)$ — energy eigenfunction; $V(x)$ — slowly varying potential; $E$ — energy eigenvalue; $m$ — mass; $\hbar$ — reduced Planck constant. The natural variable is the local classical momentum, real where the motion is allowed and imaginary where it is forbidden.

p(x)=\sqrt{2m\,[E-V(x)]} \qquad (2)

where: $p(x)$ — real in the classically allowed region $E>V$ ; imaginary where $E<V$ . In an allowed region the WKB solution is a superposition of right- and left-moving locally-plane waves, with the amplitude set by the momentum.

\psi(x)\approx \frac{C}{\sqrt{p(x)}}\exp\!\left[\pm\frac{i}{\hbar}\int^x p(x')\,\mathrm{d}x'\right] \qquad (3)

where: $C$ — normalization constant; $\pm$ — right- and left-moving components. Physical read: the phase is the classical action in units of $\hbar$ , and $|\psi|^2\propto1/p$ says the probability tracks the classical dwell time. In a forbidden region the same form holds with $p$ replaced by its magnitude, giving real exponential growth or decay — the seed of tunneling.

\psi(x)\approx \frac{C}{\sqrt{|p(x)|}}\exp\!\left[\pm\frac{1}{\hbar}\int^x |p(x')|\,\mathrm{d}x'\right] \qquad (4)

where: $|p|=\sqrt{2m(V-E)}$ , real where $E<V$ . The decaying branch is the one this article hands to the tunneling article. The method is trustworthy only when the local de Broglie wavelength changes slowly on its own scale — a condition we will derive, not merely quote.

\left|\frac{\mathrm{d}\lambda}{\mathrm{d}x}\right|\ll 1,\qquad \lambda(x)=\frac{2\pi\hbar}{p(x)} \qquad (5)

where: $\lambda(x)$ — local de Broglie wavelength, drawn from the de Broglie relation $\lambda=h/p$ .

Derivation AThe WKB wavefunction from the ħ-expansion of the phase

The goal is to obtain both the phase and the amplitude of the WKB wavefunction from a single controlled procedure — substituting the exponential ansatz and organizing the resulting nonlinear equation as an asymptotic series in the small parameter. The leading order must reproduce classical mechanics (the Hamilton–Jacobi action) and the next order must fix the amplitude so that probability flux is conserved. Nothing here is postulated; both pieces fall out order by order, and each coefficient equation is confirmed symbolically.

Result. $\psi(x)\approx p^{-1/2}\exp[\pm\tfrac{i}{\hbar}\int^x p\,\mathrm{d}x']$ .

Step 1. Substitute $\psi=e^{iS/\hbar}$ and compute its derivatives: $\psi'=\tfrac{i}{\hbar}S'\psi$ and $\psi''=[\tfrac{i}{\hbar}S''-\tfrac{1}{\hbar^2}(S')^2]\psi$ . Why: the exponential converts derivatives of the wavefunction into derivatives of the phase, the object we will expand.

Step 2. Insert into the Schrödinger equation, cancel the common factor, and rearrange into the exact Riccati form.

(S')^2 - i\hbar S'' = 2m(E-V)=p^2

Why: this is exact — no approximation yet. The nonlinearity $(S')^2$ is the price of the substitution; the $i\hbar S''$ term is the sole quantum correction.

Step 3. Expand $S=S_0+\hbar S_1+\hbar^2 S_2+\cdots$ and collect powers: order $\hbar^0$ gives $(S_0')^2=p^2$ ; order $\hbar^1$ gives $2S_0'S_1'-iS_0''=0$ . Why: treating the reduced Planck constant as the small parameter separates the classical skeleton from the leading quantum dressing. Both coefficient equations are verified symbolically.

Step 4. Solve the leading order — the phase is the classical action.

(S_0')^2=p^2\ \Rightarrow\ S_0=\pm\int p\,\mathrm{d}x

Why: this is exactly the classical Hamilton–Jacobi equation, so WKB crests track the classical momentum.

Step 5. Solve the next order for the amplitude.

2S_0'S_1'-iS_0''=0\ \Rightarrow\ S_1=\frac{i}{2}\ln p\ \Rightarrow\ e^{iS_1}=\frac{1}{\sqrt{p}}

Why: the amplitude is forced, not chosen. Reassembling $\psi\propto p^{-1/2}\exp(\pm\tfrac{i}{\hbar}\int p)$ gives a flux $j=(\hbar/m)\,\mathrm{Im}(\psi^*\psi')=|C|^2/m$ independent of position — exactly what conservation of probability current demands. *Verified: verify_aq-qm-wkb-approximation.py collects the $\hbar^0$ and $\hbar^1$ coefficients of the Riccati equation symbolically and confirms $e^{iS_1}=p^{-1/2}$ ; the constant flux $|\psi|^2(p/m)=C^2/m$ is checked algebraically.*

Go deeper · Derivation B · 1 minThe validity condition and why it always fails at a turning point

Derivation BThe validity condition and why it always fails at a turning point

This is the structural result the base treats only in passing: a quantitative criterion for when the WKB series is trustworthy, derived from the internal consistency of the expansion, together with the proof that it is guaranteed to fail at every classical turning point. The aim is not merely to state the slowly-varying condition but to show it is the same statement as "the next term in the series is negligible," and then to show the momentum vanishing at a turning point breaks it unconditionally.

Result. $|\hbar p'/p^2|\ll 1 \iff |\mathrm{d}\lambda/\mathrm{d}x|\ll1$ , and this fails as $p\to0$ .

Step 1. Demand the neglected term be small — the expansion is good when $|\hbar S_1'|\ll|S_0'|$ .

\frac{\hbar|p'|}{2p}\ \ll\ p

Why: the kept term is $S_0$ and the first approximated-small term is $\hbar S_1$ ; the asymptotic series is trustworthy precisely when that first correction is subdominant.

Step 2. Rearrange into a dimensionless ratio.

\frac{\hbar|p'|}{2p^2}\ll1

Why: this is a pure number — the fractional change of momentum over a reduced wavelength $\hbar/p$ , the honest WKB validity parameter.

Step 3. Rewrite via the wavelength $\lambda=2\pi\hbar/p$ .

\frac{\mathrm{d}\lambda}{\mathrm{d}x}=-\frac{2\pi\hbar\,p'}{p^2}\ \Rightarrow\ \left|\frac{\mathrm{d}\lambda}{\mathrm{d}x}\right|=2\pi\cdot\frac{\hbar|p'|}{p^2}

Why: up to the factor $2\pi$ the validity ratio IS the fractional change of wavelength per unit length — the intuitive picture and the formal criterion coincide.

Step 4. Evaluate at a turning point $x_0$ where $E=V(x_0)$ , so $p(x_0)=0$ but $p'(x_0)\neq0$ (a generic linear crossing).

\frac{\hbar|p'|}{p^2}\ \longrightarrow\ \infty \quad\text{as } p\to0

Why: the denominator vanishes while the numerator stays finite, so the ratio diverges no matter how smooth the potential is, and the amplitude $p^{-1/2}$ diverges in lockstep. WKB cannot be valid at a turning point — the repair is a local one, handed to the connection-formula article by name. *Verified: verify_aq-qm-wkb-approximation.py computes $\hbar|p'|/p^2$ for a harmonic well at $x=a/2$ (value $0.070$ ) and at $x=0.999a$ (value $\approx 1016$ ), exhibiting the blow-up, and confirms $\mathrm{Ai}(0)=0.3550$ is finite — the linearized problem the neighbour solves stays regular where WKB does not.*

Limiting cases

WKB is one face of the semiclassical limit; its regimes are set by $\hbar$ , the smoothness of $V$ , and proximity to turning points.

Case	Result	Meaning
Deep semiclassical (large $n$ )	$\oint p\,\mathrm{d}x\gg\hbar$	Many wavelengths fit in the well; the validity ratio is tiny in the bulk and WKB energies become asymptotically exact (high Rydberg and vibrational states).
Near a turning point	$\lvert E-V(x)\rvert\to0$	The amplitude $p^{-1/2}$ diverges; replace by the exact Airy solution of the linearized potential and stitch with the connection formulae (their own article).
Classically allowed, $E>V$	$p$ real	Oscillatory wave $p^{-1/2}\exp(\pm\tfrac{i}{\hbar}\int p)$ ; probability $\propto 1/p$ piles up where the particle is slow.
Classically forbidden, $E<V$	$p$ imaginary	Real exponential growth/decay $\lvert p\rvert^{-1/2}\exp(\pm\tfrac1\hbar\int\lvert p\rvert)$ — the decaying branch is the tunneling amplitude, cashed by the Gamow article.

Worked examples

Example 1 — classical dwell-time probability on a linear ramp. Take a particle of mass $m=1$ and energy $E=1$ on the ramp $V(x)=Fx$ with $F=0.5$ (natural units), so the turning point is at $x_t=E/F=2$ . Read off where WKB says the particle is most likely found, and by how much.

Step 1. Write the local momentum: $p(x)=\sqrt{2m(E-V)}=\sqrt{2(1-0.5x)}$ .

Step 2. Evaluate at two points, one deep in the allowed region and one near the turning point: $p(0.2)=\sqrt{2(0.9)}=1.342$ and $p(1.6)=\sqrt{2(0.2)}=0.632$ .

Step 3. The WKB probability density is $|\psi|^2\propto1/p$ , so the ratio of densities is $\tfrac{1/p(1.6)}{1/p(0.2)}=\tfrac{p(0.2)}{p(1.6)}$ .

Step 4. Compute: the particle is $2.12\times$ more likely to be found at $x=1.6$ than at $x=0.2$ — precisely because it moves slower near the turning point, spending more time there. This is the classical dwell-time law $P\propto1/v\propto1/p$ emerging from the quantum amplitude.

Example 2 — the validity ratio blows up at a turning point. Take a harmonic well $V=\tfrac12 m\omega^2 x^2$ (units $m=\omega=\hbar=1$ ) at the $n=5$ energy $E=(5+\tfrac12)=5.5$ , turning point $a=\sqrt{2E}=3.317$ . Check WKB's own trustworthiness in the bulk versus at the edge.

Step 1. The momentum is $p(x)=\sqrt{2E-x^2}$ and its derivative $p'=-x/p$ , so the validity ratio is $r(x)=\hbar|p'|/p^2=|x|/p^3$ .

Step 2. Deep in the well, at $x=a/2=1.66$ : $p=\sqrt{11-2.75}=2.87$ , giving $r=1.66/2.87^3=\mathbf{0.070}$ — comfortably $\ll1$ , WKB is excellent here.

Step 3. Near the edge, at $x=0.999a=3.31$ : $p=\sqrt{11-10.98}=0.148$ , giving $r=3.31/0.148^3\approx\mathbf{1016}$ — utterly $\gg1$ .

Step 4. The ratio has grown by a factor of about $1.5\times10^4$ over the same well. WKB is trustworthy in the middle and invalid at the edge, exactly as the derivation predicts — and exactly why the turning-point repair is a separate, unavoidable step.

Physicist

Origins · 1 minWKB Approximation and how this was discovered

WKB Approximation

1923–1926 · Zürich, Copenhagen, Cambridge

The method carries three names from a single year. In 1926 Gregor Wentzel (Zürich), Hendrik Kramers (Copenhagen), and Léon Brillouin (Paris) independently published the $\hbar$ -expansion of the Schrödinger equation and the turning-point connection formulae — hence "WKB." The mathematics was older: Harold Jeffreys had developed the asymptotic method for second-order differential equations in 1923 (British texts write "JWKB"), and the phase-integral idea traces back through Liouville and Green in the 1830s. The physics that the 1926 trio added was the interpretation — that the leading phase is the classical action and the $p^{-1/2}$ amplitude is the semiclassical probability — turning a mathematical asymptotic trick into the bridge between wave mechanics and classical trajectories.

History

1837

— Liouville and Green develop asymptotic solutions of second-order ODEs (the phase-integral method)

1923

— Harold Jeffreys gives the general asymptotic treatment of linear second-order equations

1926

— Wentzel, Kramers, and Brillouin independently apply the ħ-expansion and connection formulae to the Schrödinger equation

1926

— the method is used to rederive the Bohr–Sommerfeld quantization rule within wave mechanics

1928

— Gamow, and independently Gurney and Condon, apply the forbidden-region form to alpha-decay tunneling

Going further

References. Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., Ch. 9 · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., Sec. 2.5/Appendix · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 16 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. 1, Complement H_XI · Townsend, A Modern Approach to Quantum Mechanics, Ch. 10 · Bender & Orszag, Advanced Mathematical Methods, Ch. 10. This article owns the WKB wavefunction, its $p^{-1/2}$ amplitude, and the validity condition. The quantization condition $\oint p\,\mathrm{d}x=(n+\tfrac12)h$ is set up here and derived in WKB Quantization and the Bohr–Sommerfeld Condition; the phase-integral rule and Maslov ledger themselves belong to Bohr–Sommerfeld and Semiclassical Quantization in 1D, which deferred the wavefunction machinery here by name. The Airy patch and connection formulae are WKB Connection Formulae; the forbidden-region decaying branch is cashed in WKB Tunneling and the Gamow Factor. The deep validity-and-breakdown analysis and higher-order $\hbar$ corrections are Validity and Breakdown of WKB, and the graduate-register treatment is The WKB Approximation at QM11 level, which extends this article rather than repeating it.

Provenance. Equations follow the standard treatments in Griffiths & Schroeter Ch. 9, Sakurai Sec. 2.5, Shankar Ch. 16, and Bender & Orszag Ch. 10. A verifier should: (1) reproduce the Riccati equation and the $\hbar^0,\hbar^1$ orders giving $S_0=\int p\,\mathrm{d}x$ and amplitude $p^{-1/2}$ ; (2) check the harmonic-oscillator integral $\int\sqrt{2mE-m^2\omega^2x^2}\,\mathrm{d}x=\pi E/\omega$ yielding $E_n=(n+\tfrac12)\hbar\omega$ ; (3) confirm the connection-formula $\pi/4$ phases and the Maslov $+\tfrac12$ ; (4) verify the Gamow exponent and its rectangular-barrier limit. The hydrogen value $13.6\,\mathrm{eV}$ is the standard Rydberg energy.

Builds on: Time-Independent Schrödinger Equation · Classical Hamilton–Jacobi Theory · Airy Functions · de Broglie Matter Waves Leads to: Non-Degenerate Time-Independent Perturbation Theory · Fine Structure of Hydrogen · Stationary Phase and the Classical Limit

Is the WKB series convergent?

No — it is asymptotic. The terms typically shrink for a while and then grow, so there is an optimal truncation (usually at the first or second term). Keeping "more" terms past that point makes the answer worse, not better. This is why WKB is stated as a leading-order semiclassical result rather than an exact expansion.

Why write

\psi=e^{iS/\hbar}

rather than expanding

\psi

itself in

\hbar

Because the physics lives in the phase, which is large (of order action/ $\hbar\gg1$ ) even when the wavefunction is smooth. Expanding $\psi$ directly would try to Taylor-expand a rapidly oscillating function in the small parameter that controls its oscillation — hopeless. Putting $\hbar$ in the exponent isolates the slowly-varying object $S$ and turns the problem into a well-behaved series for the action.

What exactly is "slowly varying" — the potential, or the wavelength?

The wavelength. A potential can be steep and WKB still work, provided the resulting de Broglie wavelength changes little over its own length. The precise criterion is $\hbar|p'|/p^2\ll1$ . Because $p$ depends on both $E$ and $V$ , the same potential can be "slowly varying" for a fast (high-energy) particle and "rapidly varying" for a slow one.

Does the amplitude

p^{-1/2}

mean the probability is infinite at a turning point?

No. The $p^{-1/2}$ divergence is an artefact of WKB breaking down, not a real infinity. The true wavefunction (the Airy function of the linearized problem) is perfectly finite at the turning point. The divergence is precisely the signal that you must switch to the exact local solution there.

Is WKB only for one dimension?

The clean form here is 1D (or a separable multidimensional problem reduced to 1D radial/angular pieces). Genuine multidimensional WKB requires quantizing invariant tori (EBK), and for classically chaotic systems the trajectory-sum trace formula. Those are graduate topics; the 1D case owns the core machinery they all build on.

How is WKB related to the classical limit

\hbar\to0

Directly: as $\hbar\to0$ the leading phase $S_0/\hbar$ dominates and the stationary-phase points of $\int\psi$ pick out trajectories obeying $\delta S_0=0$ — Hamilton's principle. WKB is the first correction beyond the strict classical limit, the term that remembers there is a wave at all (the $p^{-1/2}$ amplitude and the Maslov phases).