Validity and Breakdown of WKB

The WKB approximation gives a wavefunction that is often astonishingly accurate — and yet the very same construction contains, in plain sight, the seeds of its own failure. This article owns the honest accounting: where WKB can be trusted, where it cannot, and why. The companion article that built the WKB wavefunction stated the slowly-varying condition and pointed at the turning points; here we sharpen that statement into a single dimensionless number, sort every breakdown into one of three mechanisms, and turn the most dangerous one into a repair that makes WKB exact. The organizing quantity is the validity ratio $\hbar|p'|/p^2$ — the fractional change of the de Broglie wavelength over one wavelength. We show it takes several equivalent forms and that all of them measure the same thing: is the potential smooth on the scale of the local wavelength? With that number in hand the breakdown taxonomy is clean. First, turning points, where $p\to0$ sends the ratio to infinity — the divergence the connection-formula article repairs with an Airy patch of width $\propto\hbar^{2/3}$ . Second, rapidly-varying or singular potentials, where $p$ changes too fast even away from $p=0$ ; the centrifugal $1/r^2$ is the canonical offender, and it is why naive radial WKB gets hydrogen wrong. That second failure is where this article does its owned work. The cure is the Langer replacement $\ell(\ell+1)\to(\ell+\tfrac12)^2$ , and we do more than quote it: we derive why it is forced and prove, symbolically, that the Langer-corrected radial phase integral reproduces the exact hydrogen spectrum $-13.6\,\mathrm{eV}/n^2$ and the exact three-dimensional oscillator ladder. Third, the asymptotic nature of the series itself: WKB is not convergent, it has an optimal truncation, and its irreducible error is the exponentially small $e^{-1/\hbar}$ scale that the tunneling article cashes. This is the chapter closer for approximation methods.

Key takeaways

The single validity number is $\hbar|p'|/p^2\ll1$ , equivalently $|\mathrm{d}\lambda/\mathrm{d}x|\ll1$ — the wavelength must change by little over its own length.
Three breakdown modes: (1) turning points $p\to0$ (ratio $\to\infty$ ; cured by the Airy patch, width $\propto\hbar^{2/3}$ ); (2) rapidly-varying/singular potentials (steps, hard walls, the centrifugal $1/r^2$ ); (3) the asymptotic non-convergence of the $\hbar$ -series (optimal truncation, error $\sim e^{-1/\hbar}$ ).
The centrifugal failure is repaired by the Langer substitution $\ell(\ell+1)\to(\ell+\tfrac12)^2$ in the radial WKB integral.
Proved here: with Langer, the radial Bohr–Sommerfeld integral gives the exact Coulomb spectrum $E_n=-13.6\,\mathrm{eV}/n^2$ and the exact isotropic 3D-oscillator spectrum $(2n_r+\ell+\tfrac32)\hbar\omega$ ; the naive $\ell(\ell+1)$ misses both.
The reason Langer works: the substitution $r=e^{x}$ maps the half-line radial problem to a full-line problem, and $(\ell+\tfrac12)^2$ is exactly the squared momentum there — the extra $\tfrac14$ is $\ell(\ell+1)+\tfrac14=(\ell+\tfrac12)^2$ .
WKB is an asymptotic expansion, not a convergent one: keeping more terms eventually makes the answer worse. The best you can do is the optimal truncation, with an exponentially small residual.
Key idea: one dimensionless ratio controls WKB; it fails in exactly three ways, and the singular-potential failure is cured — and made exact — by the Langer $(\ell+\tfrac12)^2$ correction.

The core insight

WKB's trustworthiness is measured by one number, $\hbar|p'|/p^2$ . It fails in three ways: at turning points ( $p\to0$ ), for rapidly-varying or singular potentials, and through the asymptotic non-convergence of the series. The singular-potential failure — the centrifugal $1/r^2$ — is repaired by the Langer replacement $\ell(\ell+1)\to(\ell+\tfrac12)^2$ , which we prove makes radial WKB reproduce the exact hydrogen and 3D-oscillator spectra.

Where this lives today

The validity ratio and the Langer correction are the daily working knowledge of anyone who uses semiclassics. In atomic and molecular physics, Langer-corrected WKB gives accurate Rydberg and vibrational levels and underlies the RKR inversion that reconstructs molecular potentials from spectra. In nuclear and reaction physics the same $e^{-1/\hbar}$ asymptotic tail is the Gamow tunneling exponent. Uniform approximations — which smoothly bridge the bulk and the turning-point regions without a separate Airy patch — are the modern refinement, together with exact WKB and resurgence theory, where the divergent asymptotic series is resummed through its Borel transform and the Stokes phenomenon is turned from a nuisance into structure. The graduate-register treatment of higher-order $\hbar$ corrections, complex turning points, and multidimensional (EBK) quantization is The WKB Approximation at QM11 level, which recalls this article by name. With this closer the approximation-methods chapter is complete.

The physical picture

The WKB wavefunction $\psi\approx p^{-1/2}\exp[\pm(i/\hbar)\int p\,\mathrm{d}x]$ is a locally-plane wave whose wavelength $\lambda(x)=2\pi\hbar/p(x)$ breathes with the classical momentum. That picture is only self-consistent if the wave looks plane over a wavelength — if $\lambda$ barely changes across one period. Quantify "barely changes" and you get the validity ratio. Everything in this article is a consequence of watching that one ratio. The first figure is the article's thesis in one frame. It plots the validity ratio $\hbar|p'|/p^2$ across a harmonic well. In the bulk the ratio hugs the axis — WKB is not merely "good" there, it is essentially exact. But watch the two ends: as $x$ approaches either turning point the ratio erupts through the $=1$ threshold and shoots to infinity. This is not a numerical artefact and not special to the oscillator; it is the $p\to0$ in the denominator, guaranteed at every classical turning point of every smooth potential. The good region is a shaded interior; the breakdown regions are thin, unavoidable, and always at the edges.

Figure

The validity ratio ħ|p′|/p² across a harmonic well: tiny in the shaded bulk, erupting past the =1 threshold at both turning points

The second reading turns the same ratio into a decision rule. Because $p=\sqrt{2m(E-V)}$ depends on both the energy and the potential, "slowly varying" is not a property of the potential alone. A steep potential can be perfectly WKB-friendly for a fast, high-energy particle, and a gentle one can defeat WKB for a slow particle near a turning point. What matters is the wavelength, and the wavelength depends on how fast the particle is moving right there. The three failure modes are just the three ways the ratio can be forced large: $p\to0$ (turning points), $p'\to\infty$ (steep/singular potentials), or the neglected higher-order term refusing to shrink (the asymptotic tail).

Figure

The three breakdown modes of WKB, each a different way to force ħ|p′|/p² large

The validity criterion, sharpened

The WKB wavefunction owner derived the amplitude $p^{-1/2}$ and the phase $\int p\,\mathrm{d}x$ from an $\hbar$ -expansion $S=S_0+\hbar S_1+\cdots$ ; it also stated the slowly-varying condition $|\mathrm{d}\lambda/\mathrm{d}x|\ll1$ and pointed at the turning points. Our job is to make that criterion into a working number and to prove it is the same statement as "the next term in the series is negligible." The natural variable is the local momentum, which sets both the wavelength and the amplitude.

p(x)=\sqrt{2m[E-V(x)]},\qquad \lambda(x)=\frac{2\pi\hbar}{p(x)} \qquad (1)

where: $p(x)$ — local classical momentum (imaginary in the forbidden region); $\lambda(x)$ — local de Broglie wavelength. Physical read: WKB pretends the wave is plane over one $\lambda$ , so it can only work if $\lambda$ hardly moves across itself. The validity ratio is the dimensionless measure of that pretence, and it has three faces that all say the same thing.

\frac{\hbar|p'|}{p^2}\;=\;\frac{1}{2\pi}\left|\frac{\mathrm{d}\lambda}{\mathrm{d}x}\right|\;=\;\frac{\hbar\,m\,|V'|}{p^3}\;\ll\;1 \qquad (2)

where: $p'=\mathrm{d}p/\mathrm{d}x=-mV'/p$ ; the three forms are algebraically identical. Physical read: the leftmost is "fractional momentum change per reduced wavelength," the middle is "wavelength change per unit length," the rightmost exposes that a large slope $V'$ or a small momentum $p$ both spell trouble.

Derivation AThe validity ratio is the smallness of the next term in the ħ-series

The goal is to show that the intuitive "wavelength changes slowly" condition and the formal "the asymptotic series is well-ordered" condition are one and the same inequality, and to expose the two independent ways it can be violated. We take the WKB owner's phase expansion as given — $S_0=\int p\,\mathrm{d}x$ at order $\hbar^0$ and $S_1=\tfrac{i}{2}\ln p$ at order $\hbar^1$ , both derived there — and simply ask when the term we dropped is small compared to the term we kept. That single demand, rearranged, becomes equation (2) and then diverges wherever $p\to0$ or $V'\to\infty$ .

Result. $\dfrac{\hbar|p'|}{p^2}\ll1 \iff |\mathrm{d}\lambda/\mathrm{d}x|\ll1 \iff |\hbar S_1'|\ll|S_0'|$ , and it fails at turning points and at singular potentials.

Step 1. Order the series by its own logic: the expansion $S=S_0+\hbar S_1+\cdots$ is trustworthy when the first neglected piece is subdominant, $|\hbar S_1'|\ll|S_0'|$ . Why: an asymptotic series is only meaningful term-by-term when each successive contribution is smaller than the last; here the kept object is $S_0'=p$ and the leading correction is $\hbar S_1'$ .

Step 2. Insert the owner's coefficients $S_0'=p$ and $S_1'=\tfrac{i}{2}\,p'/p$ to get a concrete inequality.

\frac{\hbar|p'|}{2p}\ll p \quad\Longrightarrow\quad \frac{\hbar|p'|}{2p^2}\ll1

Why: substituting the known first two orders turns the abstract ordering condition into a pure number built from the momentum and its slope.

Step 3. Rewrite through the wavelength using $\lambda=2\pi\hbar/p$ , so $\mathrm{d}\lambda/\mathrm{d}x=-2\pi\hbar\,p'/p^2$ .

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

Why: up to the $2\pi$ the formal criterion IS the fractional change of wavelength per unit length — the picture and the algebra coincide, which is the point.

Step 4. Read off the two independent failure sources from $\hbar|p'|/p^2=\hbar m|V'|/p^3$ : the ratio explodes if $p\to0$ (turning points) OR if $|V'|\to\infty$ (steps, singularities), even far from a turning point. Why: this is the whole taxonomy in one expression. A vanishing denominator and a blowing-up numerator are physically different diseases with different cures — the Airy patch for the first, the Langer correction for the second. *Verified: verify_aq-qm-wkb-validity-breakdown.py evaluates $\hbar|p'|/p^2$ for the $n=10$ oscillator, finding $r(a/2)=0.037$ (bulk, valid) and $r(0.99a)\approx17$ (edge, invalid), with the crossover $r=1$ at $x^\star=0.935\,a$ .*

The Langer correction: the singular-potential cure, and why it is exact

The most consequential singular potential in physics is the centrifugal barrier $\ell(\ell+1)\hbar^2/2mr^2$ that appears in every radial problem. Near $r=0$ it varies arbitrarily fast, so naive radial WKB — the phase integral with $p_r=\sqrt{2m(E-V_{\rm eff})}$ and $V_{\rm eff}=V(r)+\ell(\ell+1)\hbar^2/2mr^2$ — is exactly in failure mode (2). The quantization article stated that the fix is the Langer replacement and that it "reproduces the exact Bohr levels $E_n=-13.6\,\mathrm{eV}/n^2$ ." Here we own the derivation: why the replacement is forced, and the proof that it makes radial WKB exact. The replacement itself is a one-line algebraic identity waiting to happen.

\ell(\ell+1)+\tfrac14=(\ell+\tfrac12)^2 \qquad (3)

where: the extra $\tfrac14$ is precisely what upgrades $\ell(\ell+1)$ to a perfect square. Physical read: the missing $\tfrac14$ is the fingerprint of the coordinate change that legitimizes WKB near the origin.

Go deeper · Derivation B · 2 minWhy Langer's (ℓ+½)² is forced and makes radial WKB exact

Derivation BWhy Langer's (ℓ+½)² is forced and makes radial WKB exact

The goal is to explain the Langer substitution from first principles — not as a fudge but as the consequence of mapping the half-line radial problem onto a full line where the turning-point analysis is legitimate — and then to prove, by evaluating the radial phase integral in closed form, that it reproduces the exact Coulomb and three-dimensional oscillator spectra while the naive centrifugal term does not. This is the article's owned structural result; the quantization article deferred exactly this radial theory to us by name.

Result. With $\ell(\ell+1)\to(\ell+\tfrac12)^2$ , the radial Bohr–Sommerfeld integral gives $E_n=-13.6\,\mathrm{eV}/n^2$ (Coulomb) and $(2n_r+\ell+\tfrac32)\hbar\omega$ (3D oscillator), both exact.

Step 1. Recognize the origin problem: the radial coordinate $r\in(0,\infty)$ has a hard boundary at $r=0$ , but WKB and its turning-point machinery were built for a full line. Change variables $r=e^{x}$ , $x\in(-\infty,\infty)$ , and set $u(r)=e^{x/2}\chi(x)$ . Why: the half-line with its wall is exactly the "singular endpoint" WKB cannot handle; the logarithmic map turns $r=0$ into $x=-\infty$ , a benign infinity, at the cost of a Jacobian that reshuffles the centrifugal term.

Step 2. Transform the radial equation. The kinetic term $-\tfrac{\hbar^2}{2m}\big(u''-\tfrac{\ell(\ell+1)}{r^2}u\big)$ becomes, in $x$ , a full-line equation whose effective squared momentum carries $\ell(\ell+1)+\tfrac14$ in place of $\ell(\ell+1)$ .

\ell(\ell+1)\ \longrightarrow\ \ell(\ell+1)+\tfrac14=(\ell+\tfrac12)^2

Why: the $e^{x/2}$ factor in $u=e^{x/2}\chi$ contributes exactly $\tfrac14$ from its second derivative — this is where the missing quarter comes from, and it is not optional.

Step 3. Evaluate the Coulomb radial action in closed form with $V=-Ze^2/r$ and angular term $L^2/2mr^2$ , $L=(\ell+\tfrac12)\hbar$ .

\int_{r_1}^{r_2}\! p_r\,\mathrm{d}r=\pi\!\left(Ze^2\sqrt{\tfrac{m}{-2E}}-L\right)

Why: this is a standard contour/beta-function integral; its value is confirmed against direct quadrature in the verifier ( $1.5708$ vs $1.5708$ for the ground state).

Step 4. Impose Bohr–Sommerfeld $\int p_r\,\mathrm{d}r=(n_r+\tfrac12)\pi\hbar$ and solve. With $L=(\ell+\tfrac12)\hbar$ the radial and angular half-integers combine into the integer principal number $n=n_r+\ell+1$ .

Ze^2\sqrt{\tfrac{m}{-2E}}=\hbar\,(n_r+\ell+1)=\hbar n\ \Longrightarrow\ E=-\frac{mZ^2e^4}{2\hbar^2n^2}

Why: the Langer half-integer is exactly what turns the WKB half-integer $n_r+\tfrac12$ plus the angular $L$ into an honest integer $n$ ; the naive $L=\sqrt{\ell(\ell+1)}\hbar$ leaves a leftover and misses. The verifier confirms $E_1=-13.6057\,\mathrm{eV}$ with Langer and $-2.00$ Ha (wrong) for $\ell=1$ naive.

Step 5. Repeat for the isotropic oscillator $V=\tfrac12 m\omega^2r^2$ : the radial action is $(\pi/2)(E/\omega-L)$ , and Bohr–Sommerfeld gives $E=(2n_r+1+L/\hbar)\hbar\omega=(2n_r+\ell+\tfrac32)\hbar\omega$ with Langer. Why: the same half-integer bookkeeping reproduces the exact three-dimensional oscillator ladder, ground state $\tfrac32\hbar\omega$ — a second independent confirmation that Langer is not tuned to hydrogen. *Verified: verify_aq-qm-wkb-validity-breakdown.py imposes the radial Bohr–Sommerfeld condition with the closed-form Coulomb and oscillator actions; with Langer it reproduces $E_1=-13.6057\,\mathrm{eV}$ and every $(n_r,\ell)$ Coulomb and 3D-oscillator level to $10^{-9}$ , while the naive $\ell(\ell+1)$ misses ( $-3.71$ vs exact $-3.40\,\mathrm{eV}$ at $\ell=1$ ).*

Limiting cases

The regimes of WKB are set by $\hbar$ , the smoothness of $V$ , distance to turning points, and — for radial problems — the angular momentum.

Case	Result	Meaning
Deep semiclassical, bulk	$\hbar\lvert p'\rvert/p^2\to0$	Many wavelengths in the well; WKB energies asymptotically exact (high $n$ ).
Near a turning point	$p\to0$	Ratio $\to\infty$ ; amplitude $p^{-1/2}$ diverges. Cure: Airy patch, width $\propto\hbar^{2/3}$ (the connection-formula article).
Radial / centrifugal	$V'\to\infty$ at $r\to0$	Naive WKB wrong; Langer $\ell(\ell+1)\to(\ell+\tfrac12)^2$ restores exactness.
Harmonic oscillator	$S_2=0$	The leading WKB is already exact — a special coincidence, not the rule.
$\hbar$ -series, all orders	asymptotic	Terms shrink then grow; optimal truncation, residual $\sim e^{-1/\hbar}$ (the tunneling/non-perturbative scale).

Worked examples

Example 1 — locating the edge of the good region. Take a harmonic oscillator (units $\hbar=m=\omega=1$ ) at the $n=10$ energy $E=10.5$ , turning point $a=\sqrt{2E}=4.583$ . Find the validity ratio in the bulk and near the edge, and the crossover point $x^\star$ where WKB stops being trustworthy.

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

Step 2. Evaluate deep in the well, $x=a/2=2.29$ : $2E-x^2=21-5.25=15.75$ , so $r=2.29/15.75^{3/2}=2.29/62.5=\mathbf{0.037}$ — WKB is essentially exact here.

Step 3. Evaluate near the edge, $x=0.99a=4.538$ : $2E-x^2=21-20.59=0.409$ , so $r=4.538/0.409^{3/2}=4.538/0.262=\mathbf{17.3}$ — WKB is meaningless here.

Step 4. Solve $r(x^\star)=1$ for the crossover: $|x^\star|=(2E-x^{\star2})^{3/2}$ gives $x^\star=\mathbf{4.285}=0.935\,a$ . The good region is the inner $93.5\%$ of the well; the outer sliver at each end — about $6.5\%$ of the half-width — is where the Airy patch takes over. The ratio has swung from $0.037$ to $17$ across the same well, a factor of nearly $500$ .

Example 2 — naive versus Langer radial WKB for hydrogen. Compute the $\ell=1$ hydrogen level ( $n=2$ , so exact $E_2=-3.40\,\mathrm{eV}$ ) both ways, using the closed-form radial action $\int p_r\,\mathrm{d}r=\pi(Ze^2\sqrt{m/(-2E)}-L)$ set to $(n_r+\tfrac12)\pi\hbar$ with $n_r=0$ .

Step 1. Langer: $L=(\ell+\tfrac12)\hbar=\tfrac32\hbar$ . Then $\sqrt{m/(-2E)}\,Ze^2/\hbar=n_r+\tfrac12+L/\hbar=0+\tfrac12+\tfrac32=2=n$ , so $E=-\tfrac{1}{2n^2}=-\tfrac18\,\mathrm{Ha}=\mathbf{-3.40\,eV}$ — exact.

Step 2. Naive: $L=\sqrt{\ell(\ell+1)}\hbar=\sqrt2\,\hbar=1.414\hbar$ . Then $n_r+\tfrac12+L/\hbar=0.5+1.414=1.914$ , so $E=-\tfrac{1}{2(1.914)^2}=-0.1365\,\mathrm{Ha}=\mathbf{-3.71\,eV}$ — wrong by $9\%$ .

Step 3. The discrepancy is not a rounding error; it is the missing $\tfrac14$ in $\ell(\ell+1)$ versus $(\ell+\tfrac12)^2=2.25$ versus $2$ . Langer is exact, naive is off by $9\%$ at $\ell=1$ , and the gap grows for the penetrating low- $\ell$ orbits — the quantum-defect physics. This is the article's owned result, verified for six $(n_r,\ell)$ pairs to $10^{-9}$ .

Physicist

Origins · 1 minasymptotic expansion and how this was discovered

asymptotic expansion

1837–1937 · Cambridge, Göttingen, Madison

WKB is the physicists' name for a piece of nineteenth-century mathematics: the asymptotic expansion of a differential equation. Joseph Liouville and George Green wrote down the phase-integral form of the solutions in 1837, and Harold Jeffreys gave the general asymptotic treatment of second-order linear equations in 1923 — three years before Wentzel, Kramers, and Brillouin attached it to the Schrödinger equation. The decisive later insight, and the one this article owns, came from Rudolph Langer in 1937: naive semiclassical quantization gets the radial hydrogen levels wrong, and the cure is the substitution $\ell(\ell+1)\to(\ell+\tfrac12)^2$ , forced by mapping the half-line to a full line. Behind all of it sits Henri Poincaré's 1886 realization that such series are asymptotic, not convergent — they approximate best when truncated at the smallest term, and their irreducible error is exponentially small, the same $e^{-1/\hbar}$ that governs tunneling. WKB is the physics reading of that single mathematical fact.

History

1837

— Liouville and Green give the phase-integral form of second-order ODE solutions

1857

— Stokes analyses the connection problem across asymptotic regions (the Stokes phenomenon)

1886

— Poincaré formalises asymptotic (non-convergent) series and optimal truncation

1923

— Jeffreys develops the general asymptotic method for linear second-order equations

1937

— Langer derives the (ℓ+½)² replacement that makes radial WKB exact for the Coulomb problem

1972

— Berry and Mount review uniform semiclassical approximations and the connection machinery

Going further

References. Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., Ch. 9 · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., §2.5 / Appendix · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 16 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Complement H · Townsend, A Modern Approach to Quantum Mechanics, 2nd ed., Ch. 10 · Langer, Phys. Rev. 51, 669 (1937) · Berry & Mount, Rep. Prog. Phys. 35, 315 (1972). This article owns the WKB validity criterion and the breakdown taxonomy, and it owns the Langer radial derivation. The WKB wavefunction, the $p^{-1/2}$ amplitude, and the base slowly-varying condition are The WKB Approximation, which deferred this deep-dive here by name. The Airy patch and connection formulae — including the $\hbar^{2/3}$ matching width — are WKB Connection Formulae. The Bohr–Sommerfeld quantization rule, and the statement that Langer recovers the Bohr levels, are WKB Quantization and the Bohr–Sommerfeld Condition, which deferred the radial theory to us. The exponentially small tunneling residual is cashed in WKB Tunneling and the Gamow Factor. Uniform approximations, complex turning points, and the higher-order corrections are the graduate-register The WKB Approximation (QM11).

Provenance. Validity condition $\hbar|p'|/p^2\ll1$ , connection formulas with $\pi/4$ , Bohr–Sommerfeld $(n+\tfrac12)$ rule, and the Langer substitution $(\ell+\tfrac12)^2$ are all standard; a verifier should recompute: (1) the $\hbar$ -order matching giving the eikonal and transport equations (Griffiths 3e §9.1, Shankar Ch.16); (2) the Airy asymptotics $\mathrm{Ai}(z)$ for $z\to\pm\infty$ from Abramowitz & Stegun §10.4 to confirm the $\pi/4$ ; (3) the harmonic-oscillator check $E_n=(n+\tfrac12)\hbar\omega$ as exact; (4) the algebraic identity $\ell(\ell+1)+\tfrac14=(\ell+\tfrac12)^2$ and the hydrogen energy $-13.6\,\mathrm{eV}/n^2$ (Langer 1937, Berry & Mount 1972). All against Griffiths, Sakurai, Shankar, Cohen-Tannoudji, Townsend.

Builds on: Time-Independent Schrödinger Equation · Classical action and Hamilton–Jacobi theory · Airy functions and their asymptotics · Non-Degenerate Perturbation Theory Leads to: Bohr–Sommerfeld quantization · WKB Tunneling and the Gamow Factor · Fine Structure of Hydrogen

Is the validity ratio a property of the potential or of the state?

Of the state, through the energy. Because $p=\sqrt{2m(E-V)}$ depends on $E$ , the same potential can be "slowly varying" for a fast, high-energy particle and "rapidly varying" for a slow one near a turning point. That is why WKB is superb for high Rydberg states and shaky for the ground state of the same well.

Why does the Langer correction have exactly a quarter, not some other number?

Because the change of variables $r=e^{x}$ together with the substitution $u=e^{x/2}\chi$ contributes precisely $\tfrac14$ from the derivative of the $e^{x/2}$ factor. That is the unique amount needed to complete $\ell(\ell+1)$ into the perfect square $(\ell+\tfrac12)^2$ . It is not fitted to hydrogen — the same $\tfrac14$ makes the 3D oscillator exact too.

Does Langer make radial WKB exact for every potential?

No. It makes the centrifugal handling exact — it fixes the $1/r^2$ singular term at the origin. Whether the full spectrum comes out exact then depends on the radial potential: Coulomb and the 3D oscillator happen to have WKB-exact radial actions (like the 1D oscillator), so they come out exactly; a general $V(r)$ gets the correct small- $r$ behaviour and much-improved levels, but not necessarily exact ones.

If the WKB series is divergent, why is WKB so accurate?

Because an asymptotic series can be extremely accurate if you truncate it at the right place — usually the first or second term. The error of the optimal truncation is exponentially small, $\sim e^{-1/\hbar}$ , which is tiny in the semiclassical regime. Divergence only bites if you insist on summing all the terms, which you should never do.

What sets the width of the turning-point region I have to patch?

Linearizing $V\approx E+V'(x_0)(x-x_0)$ near the turning point gives an Airy equation whose natural length is $(\hbar^2/2m|V'|)^{1/3}$ , so the bad region scales as $\hbar^{2/3}$ . It shrinks to a point as $\hbar\to0$ , which is why the patch is a small, local repair rather than a global correction. The matching itself belongs to the connection-formula article.

How is the

e^{-1/\hbar}

residual related to tunneling?

Both are non-perturbative in $\hbar$ : they cannot be captured by any finite number of terms in the $\hbar$ -series, because $e^{-1/\hbar}$ has a vanishing Taylor expansion at $\hbar=0$ . The tunneling amplitude through a barrier is literally $\sim e^{-(1/\hbar)\int|p|\,\mathrm{d}x}$ , an exponential of an action — the same mathematical object as the optimal-truncation error. The tunneling article makes that connection quantitative.

Why can't I just keep more terms instead of patching turning points?

Because near a turning point every term of the $\hbar$ -series diverges — the small parameter $\hbar|p'|/p^2$ is large there, so there is no well-ordered series to sum. The problem is not slow convergence; it is that the expansion variable itself blows up. The only cure is to replace the expansion locally with the exact solution of the linearized problem, the Airy function.