WKB Quantization and the Bohr-Sommerfeld Condition

The WKB quantization condition is the rule that turns the semiclassical wavefunction into a spectrum. Take the WKB solution inside a potential well, demand that it decay into the forbidden regions beyond both turning points, and consistency forces the action accumulated over one classical period to be quantized: $\oint p\,\mathrm{d}x=(n+\tfrac12)h$ . This is the Bohr–Sommerfeld condition — the heart of the old quantum theory — now derived, and crucially corrected by the $\tfrac12$ , from wave mechanics. The article owns that derivation. The WKB wavefunction, its $p^{-1/2}$ amplitude, and its turning-point breakdown are supplied by The WKB Approximation; the connection formulae that carry a $\pi/4$ phase across each soft turning point are supplied by WKB Connection Formulae; the raw phase-integral rule and the Maslov ledger belong to Bohr–Sommerfeld and Semiclassical Quantization in 1D. What we derive here is how those pieces combine: matching the decaying tails through both turning points, adding the two $\pi/4$ shifts, and reading off the half-integer. We prove where the $\tfrac12$ comes from — it is not Bohr's integer but the sum of two Maslov quarter-turns — and we make the counting explicit for wells with soft turning points, hard walls, and mixtures. The centerpiece is a structural theorem: WKB is exact for the harmonic oscillator. We evaluate the action integral $\int_{-a}^{a}\sqrt{2mE-m^2\omega^2x^2}\,\mathrm{d}x=\pi E/\omega$ in closed form and land on $E_n=(n+\tfrac12)\hbar\omega$ with zero error for every $n$ — a genuine coincidence of the two-soft-turning-point structure, proved symbolically. Against it we place the honest contrast: for a hard-walled box the correct Maslov count ( $\pi/2$ per wall) reproduces the exact spectrum too, but the naive soft-turning-point rule misapplied to the box is only asymptotically right, off by $75\%$ at the ground state. The condition also carries a deep reading: each quantum state occupies a phase-space cell of area $h$ . That single sentence is the semiclassical density of states, the correspondence principle ( $\Delta E\to\hbar\omega_{\mathrm{cl}}$ ), and the bridge to statistical mechanics. Tunneling — the analytic continuation into the forbidden region — is deferred by name to WKB Tunneling and the Gamow Factor.

Key takeaways

The Bohr–Sommerfeld / WKB quantization condition is $\oint p\,\mathrm{d}x=(n+\tfrac12)h$ , equivalently $\int_{x_1}^{x_2}p\,\mathrm{d}x=(n+\tfrac12)\pi\hbar$ between the two turning points.
It is derived by matching the WKB solution to decaying exponentials beyond both turning points, using the connection formulae; each soft turning point contributes a $\pi/4$ phase, and the two sum to the $\tfrac12$ (the Maslov index).
The $\tfrac12$ is the wave-mechanical correction to Bohr's integer rule $\oint p\,\mathrm{d}x=nh$ — it is Maslov bookkeeping, not an arbitrary offset.
WKB is EXACT for the harmonic oscillator: the action integral evaluates to $\pi E/\omega$ , giving $E_n=(n+\tfrac12)\hbar\omega$ for all $n$ with zero error (proved symbolically).
The Maslov count is potential-dependent: two soft turning points give $(n+\tfrac12)$ ; two hard walls give the box rule (exact); a soft-plus-hard pair gives $(n+\tfrac34)$ . Using $\tfrac12$ blindly for a box is off by $75\%$ at $n=0$ .
The condition says each state occupies a phase-space cell of area $h$ — the semiclassical density of states, and the correspondence principle $\Delta E\to\hbar\omega_{\mathrm{cl}}$ .
Key idea: match the WKB wave to decaying tails through both turning points; the two $\pi/4$ connection shifts force $\oint p\,\mathrm{d}x=(n+\tfrac12)h$ , exact for the oscillator and asymptotic in general.

The core insight

Between two classical turning points the WKB wave must connect to decaying exponentials on both sides. Applying the connection formula at each end attaches a $-\pi/4$ phase, and demanding a single consistent oscillation in between forces the accumulated action $\int_{x_1}^{x_2}p\,\mathrm{d}x$ to equal $(n+\tfrac12)\pi\hbar$ . The $\tfrac12$ is the sum of the two quarter-turns — the Maslov index — and it makes WKB exact for the harmonic oscillator.

Where this lives today

The half-integer quantization rule is the semiclassical spine of modern physics. Its immediate application is molecular spectroscopy: vibrational term values of diatomics come straight from the Bohr–Sommerfeld integral, and the RKR (Rydberg–Klein–Rees) method runs it backwards to reconstruct a potential curve from measured levels. In atomic physics the same integral with the Langer replacement $\ell(\ell+1)\to(\ell+\tfrac12)^2$ recovers the Bohr energies and explains quantum defects of penetrating states. The phase-space-cell reading — one state per area $h$ — is the semiclassical density of states behind the Thomas–Fermi atom and the classical limit of statistical mechanics. Generalized to several degrees of freedom it becomes EBK (Einstein–Brillouin–Keller) torus quantization for integrable systems and, for chaotic ones, Gutzwiller's periodic-orbit trace formula. The tunneling continuation of this same machinery is WKB Tunneling and the Gamow Factor; the raw phase-integral rule and its Maslov ledger are Bohr–Sommerfeld and Semiclassical Quantization in 1D, which this article's derivation consumes by name.

The physical picture

Picture a marble rolling in a smooth bowl. Where it moves fast (deep in the well) the de Broglie wavelength $\lambda=h/p$ is short and the wave wiggles rapidly; near the turning points the marble slows, $\lambda$ stretches, and the amplitude swells. The bound state is a standing wave: as the disturbance runs from one turning point to the other and back, it must return to itself in phase. The accumulated phase is $\oint p\,\mathrm{d}x/\hbar$ — but each reflection off a soft turning point is not a hard bounce; the connection formula shows it costs an extra $\pi/4$ . Two soft reflections per round trip cost $\pi/2$ , which is exactly the $\tfrac12$ . The cleanest way to see the rule is in phase space. A classical orbit of energy $E$ traces a closed loop in the $(x,p)$ plane, and $\oint p\,\mathrm{d}x$ is the area it encloses. The quantization condition says the allowed orbits enclose areas $(n+\tfrac12)h$ — one Planck cell per state, offset by half a cell. For the harmonic oscillator these loops are ellipses of area $\pi\,x_{\max}p_{\max}=2\pi E$ , so the levels sit at equal area intervals of $2\pi\hbar=h$ , and the $\tfrac12$ -cell offset is the zero-point energy. The figure shows the nested ellipses and the staircase of enclosed areas.

Figure

Phase-space orbits enclose (n+½)h: nested oscillator ellipses and the area staircase

There is a subtlety the figure cannot show but the derivation makes precise: the $\tfrac12$ is not universal. It is the sum of Maslov contributions, one quarter per soft turning point. A well with two soft turning points (like the oscillator) gives $\tfrac12$ ; a half-well against a hard wall gives $\tfrac34$ ; a hard-walled box gives the integer-node count. The second figure contrasts the two headline cases — the oscillator, where WKB is exact, and the box, where the soft rule misapplied is only asymptotic — plotting the fractional error against quantum number.

Figure

Where WKB is exact and where it is only asymptotic: oscillator vs the box with the wrong Maslov count

The core equations

We assemble the condition from pieces the neighbours own. The starting point is the WKB wavefunction in a classically allowed well, which oscillates with the local momentum and amplitude $p^{-1/2}$ . In an allowed region ( $E>V$ ) the general WKB solution is a two-term oscillation.

\psi(x)\approx \frac{1}{\sqrt{p(x)}}\left[A\,e^{\frac{i}{\hbar}\int^x p\,\mathrm{d}x'}+B\,e^{-\frac{i}{\hbar}\int^x p\,\mathrm{d}x'}\right] \qquad (1)

where: $A,B$ — constants fixed by the connection conditions; $p=\sqrt{2m(E-V)}$ . Physical read: a standing wave whose local wavelength tracks the classical momentum. Beyond a turning point the region is forbidden and a bound state must keep only the decaying exponential.

\psi(x)\approx \frac{D}{\sqrt{|p(x)|}}\,e^{-\frac{1}{\hbar}\int_{x_t}^{x}|p|\,\mathrm{d}x'} \qquad (2)

where: $x_t$ — a turning point; $D$ — amplitude of the decaying tail. Physical read: normalizability forbids the growing branch. Crossing a soft turning point relates the decaying tail to a cosine on the allowed side, with a definite $\pi/4$ phase — the connection formula (derived in WKB Connection Formulae, consumed here).

\frac{D}{\sqrt{|p|}}\,e^{-\frac{1}{\hbar}\int_{x}^{x_t}|p|\,\mathrm{d}x'}\ \longleftrightarrow\ \frac{2D}{\sqrt{p}}\cos\!\left(\frac{1}{\hbar}\int_{x_t}^{x}p\,\mathrm{d}x'-\frac{\pi}{4}\right) \qquad (3)

where: the factor $2$ and the $-\pi/4$ come from the Airy asymptotics; this is the piece owned by the connection-formula article. Combining the two connections gives the quantization condition, written between the turning points or as a loop.

\int_{x_1}^{x_2}p(x)\,\mathrm{d}x=\left(n+\tfrac12\right)\pi\hbar\ \Longleftrightarrow\ \oint p\,\mathrm{d}x=\left(n+\tfrac12\right)h,\quad n=0,1,2,\dots \qquad (4)

where: $x_1,x_2$ — the two turning points of the well; $n$ — the number of interior nodes; $\oint$ — one full classical period. Physical read: allowed orbits enclose $(n+\tfrac12)$ Planck cells of phase-space area.

Derivation ABohr–Sommerfeld from the two connection formulae

The goal is to obtain the quantization condition by requiring a WKB bound state to decay beyond both turning points simultaneously. We consume the connection formula (owned next door) at each end, write the interior oscillation two ways — connecting inward from the right turning point and inward from the left — and demand the two descriptions agree. Consistency of a single physical wave forces the accumulated action to a half-integer multiple of $\pi\hbar$ . The two $\pi/4$ Maslov shifts, one per soft turning point, are what produce the $\tfrac12$ .

Result. $\int_{x_1}^{x_2}p\,\mathrm{d}x=(n+\tfrac12)\pi\hbar$ .

Step 1. Beyond the right turning point $x_2$ keep only the decaying tail, then connect inward: the interior wave is $\psi\propto \tfrac{2}{\sqrt p}\cos\!\big(\tfrac1\hbar\int_x^{x_2}p\,\mathrm{d}x'-\tfrac{\pi}{4}\big)$ . Why: normalizability kills the growing exponential on the right; the connection formula attaches a $-\pi/4$ as the wave enters the allowed region.

Step 2. Beyond the left turning point $x_1$ do the same, connecting inward from the left: $\psi\propto \tfrac{2}{\sqrt p}\cos\!\big(\tfrac1\hbar\int_{x_1}^{x}p\,\mathrm{d}x'-\tfrac{\pi}{4}\big)$ . Why: the identical logic on the left produces its own $-\pi/4$ ; both turning points are soft, so both contribute a quarter-turn.

Step 3. These two cosines describe the same interior wave, so their phases may differ only by a multiple of $\pi$ (up to an overall sign).

\left[\frac{1}{\hbar}\int_{x_1}^{x}p\,\mathrm{d}x'-\frac{\pi}{4}\right]+\left[\frac{1}{\hbar}\int_{x}^{x_2}p\,\mathrm{d}x'-\frac{\pi}{4}\right]=n\pi

Why: a single-valued standing wave cannot have two inconsistent phase descriptions; the freedom is exactly $n\pi$ , which is what indexes the states.

Step 4. The $x$ -dependent parts combine into the full action integral; the two quarter-turns add to $\pi/2$ .

\frac{1}{\hbar}\int_{x_1}^{x_2}p\,\mathrm{d}x-\frac{\pi}{2}=n\pi

Why: $\int_{x_1}^{x}+\int_{x}^{x_2}=\int_{x_1}^{x_2}$ independent of the interior point $x$ ; the two $-\pi/4$ 's are the Maslov sum.

Step 5. Solve.

\int_{x_1}^{x_2}p\,\mathrm{d}x=\left(n+\tfrac12\right)\pi\hbar\ \Longleftrightarrow\ \oint p\,\mathrm{d}x=\left(n+\tfrac12\right)h

Why: the $\tfrac12$ is the sum of the two soft-turning-point Maslov quarter-turns — the wave-mechanical correction to Bohr's integer rule $\oint p\,\mathrm{d}x=nh$ . *Verified: verify_aq-qm-wkb-quantization-condition.py confirms the two $\pi/4$ shifts sum to the rational $\tfrac12$ , and that the next derivation reproduces the exact oscillator spectrum from this rule.*

Go deeper · Derivation B · 1 minWKB is exact for the harmonic oscillator

Derivation BWKB is exact for the harmonic oscillator

This is the structural theorem the article owns: not merely that WKB is good for the oscillator, but that it is exact for every level, and why. The engine is a single closed-form integral. We compute the action $\int_{-a}^{a}p\,\mathrm{d}x$ for the parabolic potential symbolically, insert it into the quantization condition, and land on the exact ladder $E_n=(n+\tfrac12)\hbar\omega$ with identically zero error — a genuine coincidence of the two-soft-turning-point structure, not an approximation that happens to be close.

Result. $E_n=(n+\tfrac12)\hbar\omega$ , exact for all $n$ .

Step 1. For $V=\tfrac12 m\omega^2 x^2$ the turning points are $x_{1,2}=\mp a$ with $a=\sqrt{2E/(m\omega^2)}$ , and $p=\sqrt{2mE-m^2\omega^2x^2}$ . Why: $E=V$ fixes the classical amplitude $a$ ; the momentum vanishes there and is real in between.

Step 2. Evaluate the action integral in closed form (substitute $x=a\sin\theta$ ).

\int_{-a}^{a}\sqrt{2mE-m^2\omega^2x^2}\,\mathrm{d}x=\frac{\pi E}{\omega}

Why: the integral is the area of a half-ellipse in $(x,p)$ ; the $m$ 's cancel and it collapses to $\pi E/\omega$ exactly — verified symbolically, not numerically.

Step 3. Impose the quantization condition (4).

\frac{\pi E}{\omega}=\left(n+\tfrac12\right)\pi\hbar

Why: the oscillator has two soft turning points, so the $(n+\tfrac12)$ form applies without modification.

Step 4. Solve for the spectrum.

E_n=\left(n+\tfrac12\right)\hbar\omega

Why: this matches the exact ladder-operator result for every $n$ , zero-point energy included. The exactness is special to the quadratic potential — for a general smooth well the higher-order $\hbar$ corrections to the phase do not vanish and WKB is only asymptotically accurate. *Verified: verify_aq-qm-wkb-quantization-condition.py evaluates $\int_{-a}^{a}p\,\mathrm{d}x$ symbolically to $\pi E/\omega$ and solves to $E_n=(n+\tfrac12)\hbar\omega$ exactly; it also checks $\mathrm{d}E/\mathrm{d}n=\hbar\omega$ (the correspondence-principle spacing).*

Limiting cases

The quantization shift depends on the boundary structure; the validity depends on how slowly the wavelength varies (owned by The WKB Approximation).

Case	Result	Meaning
Two soft turning points	$\oint p\,\mathrm{d}x=(n+\tfrac12)h$	Generic smooth well (oscillator, Morse). Maslov index $\mu=2$ , shift $\mu/4=\tfrac12$ .
One soft, one hard wall	$\oint p\,\mathrm{d}x=(n+\tfrac34)h$	Bouncing ball, half-harmonic well. Soft gives $\tfrac14$ , hard wall (node) gives $\tfrac12$ .
Two hard walls (box)	$\oint p\,\mathrm{d}x=(n+1)h$	Infinite square well. WKB with this count reproduces $E_k=k^2\pi^2\hbar^2/2mL^2$ exactly, $k=n+1$ .
Large quantum number	$\Delta E_n\to\hbar\omega_{\mathrm{cl}}(E_n)=\tfrac{2\pi\hbar}{T(E_n)}$	Level spacing equals the classical orbital frequency times $\hbar$ — the correspondence principle.
Harmonic oscillator	$E_n=(n+\tfrac12)\hbar\omega$	The action integral is exactly $\pi E/\omega$ ; WKB is exact for all $n$ .

Worked examples

Example 1 — the oscillator action integral, end to end. Compute $\int_{-a}^{a}p\,\mathrm{d}x$ for $V=\tfrac12 m\omega^2x^2$ and read off the spectrum. Work in natural units $m=\omega=\hbar=1$ and take the level $n=3$ .

Step 1. At $E=E_3$ the turning points are $a=\sqrt{2E}$ ; the momentum is $p=\sqrt{2E-x^2}$ .

Step 2. Substitute $x=a\sin\theta$ : $\int_{-a}^{a}\sqrt{2E-x^2}\,\mathrm{d}x=2E\int_{-\pi/2}^{\pi/2}\cos^2\theta\,\mathrm{d}\theta=2E\cdot\tfrac{\pi}{2}=\pi E$ . (With $\hbar=\omega=1$ this is the general $\pi E/\omega$ .)

Step 3. Impose $\pi E=(n+\tfrac12)\pi$ , i.e. $E_n=n+\tfrac12$ .

Step 4. For $n=3$ : $E_3=3.5\,\hbar\omega$ , identical to the exact ladder-operator value — no error, as the theorem promises. The action integral evaluated to $\pi E/\omega$ symbolically confirms exactness for every $n$ , not just $n=3$ .

Example 2 — the box, right way and wrong way. Compare the WKB spectrum of an infinite square well of width $L$ (natural units $m=\hbar=1$ , $L=1$ ) computed with the correct hard-wall count versus the naive soft rule, at $n=0$ and $n=9$ .

Step 1. The exact spectrum is $E_k=k^2\pi^2/2$ , $k=1,2,\dots$ . The momentum is constant, $p=\sqrt{2E}$ , so $\int_0^L p\,\mathrm{d}x=L\sqrt{2E}$ .

Step 2. Correct (two hard walls): the rule is $\int_0^L p\,\mathrm{d}x=(n+1)\pi\hbar$ , giving $E=(n+1)^2\pi^2/2$ . With $k=n+1$ this is exact for every $n$ — zero error at $n=0$ and $n=9$ alike.

Step 3. Wrong (soft $(n+\tfrac12)$ rule misapplied): $E=(n+\tfrac12)^2\pi^2/2$ . At $n=0$ this gives $\tfrac14\pi^2/2$ versus the exact $\pi^2/2$ — a $75\%$ shortfall.

Step 4. At $n=9$ the wrong rule gives $(9.5)^2$ versus the exact $10^2$ : an error of $9.75\%$ . The mistake shrinks as $n$ grows (asymptotic recovery) but never vanishes — the honest lesson that the Maslov count, not a universal $\tfrac12$ , is what makes WKB exact for the box.

Physicist

Origins · 1 minBohr-Sommerfeld Quantization and how this was discovered

Bohr-Sommerfeld Quantization

1913–1926 · Copenhagen, Munich

The condition $\oint p\,\mathrm{d}q=nh$ was the organizing principle of the old quantum theory. Bohr's 1913 hydrogen model quantized angular momentum; Sommerfeld generalized it in 1915–16 to $\oint p\,\mathrm{d}q=nh$ for each separable coordinate, quantizing elliptical Kepler orbits and — with a relativistic correction — even producing the hydrogen fine-structure formula before Dirac. The rule was empirically astonishing and conceptually opaque: no one knew why phase-space areas should be integer multiples of $h$ . Wave mechanics answered that in 1926: the WKB analysis reproduces the rule, explains it as a standing-wave condition, and corrects the integer to $n+\tfrac12$ through the Maslov phases. The old quantum theory was not so much overthrown as derived — with a half-quantum of interest.

History

1913

— Bohr quantizes angular momentum for the hydrogen atom

1915

— Sommerfeld generalizes to ∮ p dq = nh for separable coordinates

1916

— Sommerfeld's relativistic elliptical orbits reproduce the hydrogen fine-structure formula

1926

— WKB analysis rederives the rule within wave mechanics and adds the ½ Maslov correction

1958

— Maslov formalizes the index counting the phase loss at caustics and turning points

Going further

References. Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., Ch. 9 · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., Sec. 2.5 · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 16 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Complement H_III / B_XI · Townsend, A Modern Approach to Quantum Mechanics, Ch. 10. This article owns the derivation of the quantization condition and the WKB-exact-for-the-oscillator theorem. The WKB wavefunction and validity condition are The WKB Approximation; the $\pi/4$ connection formulae are WKB Connection Formulae, consumed here; the phase-integral rule and Maslov ledger (the $\nu=2/3/4$ toll and the quartic-well accuracy staircase $18.2\%\to1.3\%\to0.6\%\to0.3\%$ and the neutron-bouncer levels) are Bohr–Sommerfeld and Semiclassical Quantization in 1D, recalled by name and never re-owned. Tunneling is WKB Tunneling and the Gamow Factor; the Langer-correction radial theory and higher-order quantization corrections are deferred to the graduate register.

**Provenance.*All equations are standard semiclassical results. A verifier should: (1) confirm the $\hbar$ -expansion of $-\frac{\hbar^2}{2m}\psi''+V\psi=E\psi$ yields the eikonal $(S_0')^2=p^2$ and transport $S_1=\frac{i}{2}\ln p$ equations (Griffiths 3e Ch. 9; Shankar Ch. 16); (2) recompute the harmonic-oscillator action integral $\int_{-a}^{a}p\,\mathrm{d}x=\pi E/\omega$ giving $E_n=\hbar\omega(n+1/2)$ exactly; (3) verify the Airy-asymptotic connection formulas with the $\pi/4$ shift (Sakurai 2e Sec. 2.5; Cohen-Tannoudji Complement BXI); (4) check the Maslov-index rule (1/4 per soft turning point, 1/2 per hard wall) against the infinite square well result $E_n=(n+1)^2\pi^2\hbar^2/2mL^2$ ; (5) confirm Bohr energies $-13.6\,\mathrm{eV}/n^2$ from the radial integral with the Langer correction. Numerical constant $13.6\,\mathrm{eV}$ is the standard Rydberg energy.

Builds on: Time-Independent Schrödinger Equation · Classical Action and Hamilton–Jacobi Theory · Airy Functions · Correspondence Principle Leads to: WKB Tunneling and the Gamow Factor · Fine Structure of Hydrogen · Non-Degenerate Time-Independent Perturbation Theory

Why is it called both "Bohr–Sommerfeld" and "WKB" quantization?

They are the same rule reached from two directions. Bohr (1913) and Sommerfeld (1915–16) postulated $\oint p\,\mathrm{d}q=nh$ as the quantization axiom of the old quantum theory. WKB (1926) derives it from wave mechanics as a standing-wave/connection-formula condition — and in doing so corrects the integer to $n+\tfrac12$ via the Maslov phases. "Bohr–Sommerfeld" honours the historical postulate; "WKB quantization" honours the derivation that explained and corrected it.

Is the ½ always ½?

No — this is the most important caveat. The additive constant is $\mu/4$ where $\mu$ is the Maslov index, counting $1$ per soft turning point and $2$ per hard wall (a node). Two soft turning points give $\tfrac12$ ; a soft-plus-hard pair gives $\tfrac34$ ; two hard walls give the integer-node box count. Blindly using $\tfrac12$ for a box is off by $75\%$ at the ground state.

If WKB is approximate, how can it be exact for the oscillator?

The WKB error lives in the turning-point regions, where the potential is linearized and matched to an Airy function. For the quadratic oscillator the action integral is exactly $\pi E/\omega$ and the two $\pi/4$ phases are exactly right, so the quantization integral has no residual error. It is a genuine coincidence of the parabolic potential — the higher-order $\hbar$ corrections to the phase happen to sum to zero. No other elementary potential shares this except through similar special structure.

Why does each state occupy phase-space area h?

Differentiating $\oint p\,\mathrm{d}x=(n+\tfrac12)h$ with respect to $n$ gives $\mathrm{d}(\oint p\,\mathrm{d}x)=h$ per unit increase in $n$ : consecutive orbits enclose areas differing by exactly $h$ . So the number of states below energy $E$ is the enclosed phase-space area divided by $h$ . This is the semiclassical density of states and the origin of the "one state per Planck cell" rule that underlies statistical mechanics' classical limit.

How does WKB handle the hydrogen atom?

Apply the radial integral $\oint p_r\,\mathrm{d}r=(n_r+\tfrac12)h$ with the Langer replacement $\ell(\ell+1)\to(\ell+\tfrac12)^2$ in the centrifugal term. Without Langer the WKB energies are wrong; with it they reproduce the exact Bohr levels $E_n=-13.6\,\mathrm{eV}/n^2$ . The Langer correction fixes the $r\to0$ singularity where a naive linearization fails, and its full justification is a graduate-register topic.

Does the derivation assume both turning points are "soft"?

Yes — the two-connection-formula derivation attaches a $\pi/4$ at each linear (soft) turning point, giving the $(n+\tfrac12)$ result. If a boundary is a hard wall the wavefunction must vanish there (a node), which is a different phase condition ( $\pi/2$ contribution), and the count changes accordingly. The derivation here is the two-soft-turning-point case; the limiting-cases table lists the others.