WKB approximation and Bohr-Sommerfeld quantisation

Intuition Beginner

Most quantum problems do not solve in closed form. The hydrogen atom, the harmonic oscillator, the infinite square well — these are the rare cases where the Schrödinger equation hands you exact energies and wavefunctions. Pick almost any other potential — a hill of unusual shape, a smoothly varying well, a barrier with curved walls — and the equation refuses to close. You need an approximation.

The WKB approximation, named for Wentzel, Kramers, and Brillouin (with Jeffreys credited for an earlier independent version), is the standard tool when the potential varies slowly. The key idea is that a quantum particle with momentum $p$ has a de Broglie wavelength $\lambda =h/p$. If $V(x)$ changes very little over one wavelength, the particle behaves locally like a free particle moving with the local momentum $p(x)=\sqrt{2m(E-V(x))}$. Its wavefunction should oscillate locally like $\exp (ip(x)\cdot x/\hslash )$ with a slowly drifting amplitude.

Think of light passing through a medium of slowly varying index of refraction. Geometric optics — rays following Snell's law — works when the wavelength is much shorter than the scale of variation. Where the index changes abruptly, you need full wave optics. WKB is the quantum-mechanical version of geometric optics: where the potential varies slowly compared to the de Broglie wavelength, you can replace wave mechanics by something simpler.

The cost of the approximation is that it fails near classical turning points, the values $x=a$ where the energy line $E$ touches the potential curve $V(a)=E$. At a turning point, the local momentum $p(a)=0$, the de Broglie wavelength becomes infinite, and the slow-variation condition is violated. Around the turning point, the WKB formula blows up; you need a separate trick — matching to the exact Airy-function solution of a linear potential — to bridge the oscillatory region inside the well to the exponentially decaying region in the classically forbidden zone outside.

Once you know how to match across turning points, you can count bound states. A particle trapped between two turning points $a$ and $b$ oscillates back and forth; for the wavefunction to be single-valued, the total phase accumulated over a round trip must be a multiple of $2\pi$, minus a correction for the turning points. The result is the Bohr-Sommerfeld quantisation condition: $\oint pdq=(n+1/2)h$, where the integral is taken over one full classical period and $n=0,1,2,\dots$ labels the bound state. The $1/2$ is the Maslov correction, contributed by the two soft turning points at $1/4$ each.

That same formula was the workhorse of the old quantum theory of 1913-1925, before Schrödinger's wave equation existed. Bohr used it to derive his hydrogen energy levels; Sommerfeld extended it to elliptical orbits and got the relativistic fine structure. The old quantum theory was a patchwork, but the formula it produced is recovered exactly by WKB in the semiclassical limit. WKB is the modern home of the old quantum recipe, with its limits of validity made explicit.

Tunnelling is the other major application. A particle with energy $E$ less than the height of a barrier classically cannot cross — the region inside the barrier is forbidden. Quantum mechanics allows a small transmission probability, and WKB gives a clean formula: the transmission coefficient is roughly $T\approx \exp (-2K/\hslash )$, where $K$ is the action of the imaginary-momentum trajectory across the forbidden region. Gamow used this in 1928 to explain alpha decay, with the alpha particle tunnelling through the Coulomb barrier of the nucleus. The exponential suppression with barrier height and width is the signature of quantum tunnelling.

Visual Beginner

Inside the classically allowed region, the WKB wavefunction oscillates with a local wavelength $2\pi \hslash /p(x)$. The amplitude is $1/\sqrt{p(x)}$, so the wavefunction is large where the particle moves slowly (near the turning points, where it lingers) and small where it moves fast (near the bottom of the well, where it sweeps through quickly). Outside the turning points, the wavefunction decays exponentially, vanishing rapidly in the classically forbidden zone.

Worked example Beginner

Bohr-Sommerfeld for the harmonic oscillator. Take the one-dimensional oscillator with potential $V(x)=\frac{1}{2}m{\omega }^2{x}^2$. The classical turning points at energy $E$ are the solutions of $E=\frac{1}{2}m{\omega }^2{a}^2$, giving $a=\pm \sqrt{2E/(m{\omega }^2)}$.

Compute the action integral. Inside the well, the local momentum is $p(x)=\sqrt{2m(E-V(x))}=\sqrt{2mE-m^2{\omega }^2{x}^2}$. The action $\oint pdq$ over one full period of the classical motion is the area enclosed by the phase-space orbit. Squaring the momentum identity gives the ellipse $p^2/(2mE)+m{\omega }^2{x}^2/(2E)=1$ with semi-axes $\sqrt{2mE}$ (in the $p$ direction) and $a=\sqrt{2E/(m{\omega }^2)}$ (in the $x$ direction). The area of an ellipse with semi-axes $A$ and $B$ is $\pi AB$, so

$$\oint pdq=\pi \cdot \sqrt{2mE}\cdot \sqrt{2E/(m{\omega }^2)}=2\pi E/\omega .$$

Apply the quantisation condition. Setting $\oint pdq=2\pi \hslash (n+1/2)$:

$$\frac{2\pi E}{\omega }=2\pi \hslash \left(n+\frac{1}{2}\right)⟹E_n=\hslash \omega \left(n+\frac{1}{2}\right).$$

This is the exact harmonic-oscillator spectrum. Bohr-Sommerfeld, an approximation derived from leading-order semiclassical asymptotics, returns the correct answer to all orders. The harmonic oscillator is the canonical case where the WKB formula succeeds completely — a coincidence rooted in the quadratic nature of the potential, which is exactly the local form at every turning point.

What this tells us. The half-integer correction is essential. Without it, the spectrum would start at $E_0=0$, contradicting the zero-point energy $\frac{1}{2}\hslash \omega$ that the quantum oscillator must have on dimensional and ground-state grounds. The Maslov $1/2$ is not optional; it is dictated by the connection-formula geometry at the two turning points.

Check your understanding Beginner

Formal definition Intermediate+

Consider the one-dimensional time-independent Schrödinger equation

$$-\frac{{\hslash }^2}{2m}{\psi }^{\prime \prime }(x)+V(x)\psi (x)=E\psi (x).$$

Write the wavefunction in the form

$$\psi (x)=A(x)\exp \left(\frac{i}{\hslash }S(x)\right),$$

where $S(x)$ and $A(x)$ are real. Substituting into the Schrödinger equation and separating real and imaginary parts gives

$$(S^{\prime }{)}^2=2m(E-V(x))+{\hslash }^2\frac{A^{\prime \prime }}{A},(A^2{S}^{\prime }{)}^{\prime }=0.$$

The second equation states that $j(x)=A^2(x)S^{\prime }(x)/m$ is a constant — the probability current in the WKB ansatz, conserved as expected for a stationary state.

Expand $S$ as a formal power series in $\hslash$:

$$S(x)=S_0(x)+\hslash S_1(x)+{\hslash }^2{S}_2(x)+\cdots .$$

Leading order $({\hslash }^0)$: $(S_0^{\prime }{)}^2=2m(E-V(x))$. Setting $p(x)=\sqrt{2m(E-V(x))}$ (the classical local momentum), the solutions are

$$S_0(x)=\pm {\int }^{x}p(y)dy.$$

This is precisely the action integral of the corresponding classical orbit at energy $E$. The eikonal phase is the classical action; the $\hslash \to 0$ limit of WKB is classical mechanics. This bridge is the substance of 05.05.04 Hamilton-Jacobi theory: $S_0$ satisfies the Hamilton-Jacobi equation $(S_0^{\prime }{)}^2/(2m)+V=E$.

Next-to-leading order $({\hslash }^1)$: the next-to-leading equation determines the amplitude. Substituting $S=S_0+\hslash S_1+\cdots$ into the conservation law $A^2{S}^{\prime }=\text{const}$ and using $S_0^{\prime }=\pm p$ gives

$$A(x)=\frac{C}{\sqrt{p(x)}},$$

for some constant $C$. The amplitude is inversely proportional to the square root of the local momentum — large where the particle is slow, small where it is fast.

WKB ansatz (allowed region). Combining leading and next-to-leading orders, the WKB wavefunction in the classically allowed region $E>V(x)$ is

$${\psi }_{\mathrm{WKB}}(x)=\frac{C_{+}}{\sqrt{p(x)}}\exp \left(\frac{i}{\hslash }{\int }^{x}pdy\right)+\frac{C_{-}}{\sqrt{p(x)}}\exp \left(-\frac{i}{\hslash }{\int }^{x}pdy\right).$$

Real linear combinations give the standard $\cos$ / $\sin$ form.

WKB ansatz (forbidden region). Where $E<V(x)$, set $|p(x)|=\sqrt{2m(V(x)-E)}$. The same expansion produces decaying and growing exponentials:

$${\psi }_{\mathrm{WKB}}(x)=\frac{C_{+}}{\sqrt{|p(x)|}}\exp \left(+\frac{1}{\hslash }{\int }^x|p|dy\right)+\frac{C_{-}}{\sqrt{|p(x)|}}\exp \left(-\frac{1}{\hslash }{\int }^x|p|dy\right).$$

Validity criterion. The expansion in $\hslash$ converges when the correction term ${\hslash }^2{A}^{\prime \prime }/A$ is small compared with the leading $(S_0^{\prime }{)}^2=p^2$. A short computation shows this is equivalent to

$$\boxed{\hslash \left|\frac{dp}{dx}\right|\ll p^2,}$$

which can be rewritten as $|d{\lambda }_{\mathrm{dB}}/dx|\ll 1$ where ${\lambda }_{\mathrm{dB}}(x)=2\pi \hslash /p(x)$ is the local de Broglie wavelength: the wavelength must change slowly on its own scale. The criterion fails as $p\to 0$, that is, at the classical turning points $V(a)=E$.

Connection formulas at a turning point

Near a turning point $x=a$ where $V(a)=E$ and $V^{\prime }(a)>0$ (classically allowed region to the left of $a$, forbidden to the right), expand $V(x)-E\approx V^{\prime }(a)(x-a)$. The Schrödinger equation reduces to

$${\psi }^{\prime \prime }(x)=\frac{2m{V}^{\prime }(a)}{{\hslash }^2}(x-a)\psi (x),$$

which is the Airy equation ${\psi }^{\prime \prime }=z\psi$ in the rescaled variable $z=(x-a)/\ell$ with characteristic length $\ell =({\hslash }^2/(2m{V}^{\prime }(a)){)}^{1/3}$. The decaying-at-infinity solution is the Airy function $\mathrm{Ai}(z)$, with known asymptotic behaviour

• $z\to +\infty$ (forbidden side): $\mathrm{Ai}(z)\sim \frac{1}{2\sqrt{\pi }{z}^{1/4}}\exp \left(-\frac{2}{3}{z}^{3/2}\right)$.
• $z\to -\infty$ (allowed side): $\mathrm{Ai}(z)\sim \frac{1}{\sqrt{\pi }|z{|}^{1/4}}\sin \left(\frac{2}{3}|z{|}^{3/2}+\frac{\pi }{4}\right)$.

Translating back to the original $x$ coordinate and identifying $\frac{2}{3}|z{|}^{3/2}={\int }_x^{a}p(y)dy/\hslash$ (allowed side) and $\frac{2}{3}{z}^{3/2}={\int }_a^x|p(y)|dy/\hslash$ (forbidden side), the Airy matching gives the Langer connection formula (Langer 1937 ^{[source pending]}):

$$\boxed{\frac{C_{+}}{\sqrt{p(x)}}\cos \left(\frac{1}{\hslash }{\int }_x^{a}pdy-\frac{\pi }{4}\right)⟷\frac{C_{-}}{2\sqrt{|p(x)|}}\exp \left(\frac{1}{\hslash }{\int }_a^x|p|dy\right),}$$

reading "allowed (left of $a$) → forbidden (right of $a$)" with the matching $C_{+}=C_{-}$. The arrow runs only in one direction: from the small-decaying exponential on the forbidden side to a definite cosine combination on the allowed side. Reversing the arrow loses information, because a small admixture of the exponentially growing solution is invisible to leading order on the forbidden side but dominates exponentially after a short distance. Treating the connection as one-directional is the discipline that keeps WKB consistent.

Key derivation Intermediate+

Theorem (Bohr-Sommerfeld quantisation). Let $V(x)$ be a smooth one-dimensional potential with a single well, and let $E$ lie between the bottom and the dissociation threshold so that the classical motion at energy $E$ is bounded between two turning points $a<b$ with $V(a)=V(b)=E$ and $V^{\prime }(a)<0$, $V^{\prime }(b)>0$. Then the WKB bound-state energies $E_n$ satisfy

$$\oint pdq=2{\int }_a^{b}p(x)dx=2\pi \hslash \left(n+\frac{1}{2}\right),n=0,1,2,\dots .$$

Derivation. Apply the connection formula at each turning point and demand that the wavefunctions match in the allowed interval $(a,b)$.

Match at the left turning point $a$. To the left of $a$, the wavefunction must decay (the bound state has no support at $-\infty$):

$$\psi (x)=\frac{C}{\sqrt{|p(x)|}}\exp \left(-\frac{1}{\hslash }{\int }_x^a|p|dy\right)(x<a).$$

The Langer connection — at the left turning point the allowed region is to the right of $a$, so apply the analogue with the roles of "left" and "right" swapped — gives the form in the allowed region

$$\psi (x)=\frac{2C}{\sqrt{p(x)}}\cos \left(\frac{1}{\hslash }{\int }_a^{x}pdy-\frac{\pi }{4}\right)(a<x<b).\text{\hspace{1em}(⋆)}$$

Match at the right turning point $b$. The same logic at the right turning point, where the allowed region is to the left and the forbidden decaying solution to the right, yields

$$\psi (x)=\frac{2C^{\prime }}{\sqrt{p(x)}}\cos \left(\frac{1}{\hslash }{\int }_x^{b}pdy-\frac{\pi }{4}\right)(a<x<b).\text{\hspace{1em}(⋆⋆)}$$

Consistency. Both ($\star$) and ($\star \star$) must describe the same wavefunction in $(a,b)$. Write the argument of the cosine in ($\star \star$) using ${\int }_x^{b}pdy={\int }_a^{b}pdy-{\int }_a^{x}pdy$:

$$\psi (x)=\frac{2C^{\prime }}{\sqrt{p(x)}}\cos \left(\frac{1}{\hslash }{\int }_a^{b}pdy-\frac{1}{\hslash }{\int }_a^{x}pdy-\frac{\pi }{4}\right).$$

Setting this equal to ($\star$) and using $\cos (\alpha )=\cos (\beta )$ iff $\alpha =\pm \beta +2\pi k$, the relation

$$\frac{1}{\hslash }{\int }_a^{x}pdy-\frac{\pi }{4}=-\left(\frac{1}{\hslash }{\int }_a^{b}pdy-\frac{1}{\hslash }{\int }_a^{x}pdy-\frac{\pi }{4}\right)+n\pi$$

must hold for all $x$. The $x$-dependence cancels automatically; the $x$-independent piece yields

$$\frac{1}{\hslash }{\int }_a^{b}pdy-\frac{\pi }{2}=n\pi ,n=0,1,2,\dots ,$$

which is exactly the Bohr-Sommerfeld condition ${\int }_a^{b}pdx=\pi \hslash (n+1/2)$. Doubling to recover the full period gives $\oint pdq=2\pi \hslash (n+1/2)$. The combined $-\pi /4$ shifts from the two turning points sum to $-\pi /2$ — this is the Maslov correction appearing as the half-integer offset. $\square$

The matching constants $C$ and $C^{\prime }$ are equal up to a sign that is absorbed into the normalisation of $\psi$. Choosing the overall sign gives an $n$-dependent number of nodes in $(a,b)$, in agreement with the Sturm oscillation theorem: the $n$-th excited state has exactly $n$ nodes.

Worked example: hydrogen atom radial spectrum via Bohr-Sommerfeld

For the hydrogen atom, the central-force radial equation at fixed angular momentum $\ell$ reduces to a one-dimensional problem with effective potential

$$V_{\mathrm{eff}}(r)=-\frac{e^2}{r}+\frac{{\hslash }^2\ell (\ell +1)}{2m{r}^2},$$

with radial momentum $p_r(r)=\sqrt{2m(E-V_{\mathrm{eff}}(r))}$. The classical turning points $r_{\pm }$ at energy $E<0$ are the roots of $E=V_{\mathrm{eff}}(r)$.

The Bohr-Sommerfeld integral, with the appropriate Maslov adjustment for the radial coordinate (the soft turning point at $r_{+}$ contributes $\frac{1}{4}$; the hard reflection at $r=0$ for the centrifugal barrier contributes a different offset, conventionally $\frac{1}{2}$ for the Langer-corrected form $\ell (\ell +1)\to (\ell +1/2{)}^2$), evaluates to

$$\oint p_{r}dr=2\pi \hslash \left(n_r+\frac{1}{2}\right)⟹E_{n_r,\ell }=-\frac{m{e}^4}{2{\hslash }^2(n_r+\ell +1{)}^2}.$$

Setting $n=n_r+\ell +1$ gives the Bohr formula $E_n=-m{e}^4/(2{\hslash }^2{n}^2)$, the original 1913 result reproduced exactly by WKB with the Langer correction. The combination $n=n_r+\ell +1$ is the principal quantum number; the degeneracy ${\sum }_{\ell =0}^{n-1}(2\ell +1)=n^2$ at fixed $n$ is the "accidental" $\mathrm{SO}(4)$ degeneracy of the Kepler problem 12.06.01.

Exercises Intermediate+

Lean formalization Intermediate+

Mathlib does not yet cover semiclassical asymptotics. The closest layers are:

• Mathlib.Analysis.Asymptotics.Asymptotics: $O$- and $o$-notation for asymptotic comparison, the natural language for $\hslash \to 0$ expansions.
• Mathlib.Analysis.SpecialFunctions.Trigonometric: trigonometric and exponential infrastructure, prerequisite for an eventual Airy-function library.
• Mathlib.Analysis.ODE: smooth ordinary differential equations, on which the matched-asymptotic Airy construction would build.

There is no Mathlib definition of the WKB ansatz, no Airy function, no statement of Bohr-Sommerfeld as a self-adjoint eigenvalue identity, and no Maslov-index calculus on the Lagrangian Grassmannian. The formalisation pathway is outlined in Mathlib gap analysis. The lean_status: none reflects this gap; no Lean module ships with this unit. Tyler attests intermediate-tier correctness pending external QM reviewer.

Higher-order WKB and exact WKB Master

The leading-order WKB ansatz $\psi \sim p^{-1/2}\exp (\frac{i}{\hslash }\int pdx)$ is the first term of an asymptotic series in $\hslash$. The full expansion $S={\sum }_k{\hslash }^k{S}_k$ generates corrections at every order: $S_2$ contributes a correction to the eikonal phase, $S_3$ to the amplitude, and so on. Bender and Orszag give the explicit recursion. The series is asymptotic, not convergent: the sum of all orders, naively interpreted, diverges, but the truncated $N$-th partial sum approximates the true wavefunction with error $O({\hslash }^{N+1})$ for $\hslash$ small enough.

Exact WKB, developed by Voros, Sato-Aoki-Kawai, and Delabaere-Dillinger-Pham, addresses the divergence through resurgent analysis (Voros 1983 ^{[source pending]}). The formal WKB series carries non-perturbative information encoded in the Stokes phenomena along curves in the complex $\hslash$-plane: where the partial sums of the series cross specific lines, additional exponentially small corrections of order $\exp (-\text{action}/\hslash )$ must be added. The full exact-WKB expansion, including all non-perturbative contributions, reconstructs the exact spectrum order by order — a remarkable demonstration that the asymptotic series, suitably resummed, is not lossy.

The technical machinery — Borel-Laplace resummation, alien calculus, the bridge equation — has applications well beyond the original 1D Schrödinger setting: to anharmonic oscillators (Bender-Wu 1969 Phys. Rev. 184), to instanton counting in gauge theories (Dunne-Ünsal 2014 PRD 89), to the quantum monodromy of integrable systems (Iwaki-Nakanishi 2014 J. Phys. A 47), to the Painlevé equations (Aoki-Kawai-Takei 1996 Comm. Math. Phys. 181). The intuition is that the perturbation series alone is incomplete; the missing piece is the non-perturbative contribution from tunnelling and instanton sectors, which exact WKB makes precise.

Einstein-Brillouin-Keller and the Maslov index Master

Bohr-Sommerfeld generalises to higher-dimensional integrable systems through Einstein-Brillouin-Keller (EBK) quantisation (Keller 1958 ^{[source pending]}). For an integrable Hamiltonian system on a $2n$-dimensional phase space with $n$ independent conserved quantities, Liouville-Arnold gives action-angle coordinates $(I_i,{\theta }_i)$ in which trajectories wind on $n$-dimensional tori $T^n\subset T^{\ast }M$. Quantising the action on each independent loop ${\gamma }_i$ of the torus:

$${\oint }_{{\gamma }_i}pdq=2\pi \hslash \left(n_i+\frac{{\mu }_i}{4}\right),n_i=0,1,2,\dots ,i=1,\dots ,n.$$

The integer ${\mu }_i$ is the Maslov index of the loop ${\gamma }_i$ — the winding number of its image in the Lagrangian Grassmannian $\Lambda (n)=U(n)/O(n)$ under the canonical map. Soft turning points contribute $+1$ to the Maslov count; hard reflections contribute $+2$; the total ${\mu }_i/4$ is the half-integer offset analogous to the 1D Bohr-Sommerfeld correction.

The two-dimensional rectangular box (Exercise 10) and the hydrogen atom in parabolic coordinates illustrate the EBK construction directly: the conserved quantities are the Cartesian momenta (box) or the parabolic separation constants (hydrogen). For non-separable but still integrable systems — the Toda lattice, the spherical pendulum — EBK works in any system of action-angle coordinates, and the spectrum can be computed without solving the Schrödinger equation explicitly.

For non-integrable systems the EBK quantisation breaks down. The phase-space tori dissolve (KAM theorem), trajectories become chaotic, and there is no canonical action variable to quantise. The successor framework is the Gutzwiller trace formula (Gutzwiller 1971 J. Math. Phys. 12 343), which expresses the semiclassical density of states as a sum over classical periodic orbits weighted by stability and Maslov phase. Gutzwiller is the right tool for the helium atom three-body problem, for the stadium and Sinai billiards, and for the level statistics that connect quantum chaos to random-matrix theory.

The Maslov index itself is a topological invariant of the Lagrangian Grassmannian, computed via the universal ${\det }^2:\Lambda (n)\to S^1$ map. Its integer values reflect the geometry of caustics in the underlying classical configuration; the half-integer offset in Bohr-Sommerfeld is the spectral fingerprint of this topology. The full Lagrangian-Grassmannian story sits in 05.08.03 symplectic geometry and threads through Floer theory, the Conley-Zehnder index, and the Atiyah-Singer index theorem.

Connections Master

• 05.05.04 Hamilton-Jacobi theory. The eikonal phase $S_0(x)=\int pdy$ satisfies the Hamilton-Jacobi equation $(\partial S_0/\partial x{)}^2/(2m)+V(x)=E$. The leading-order WKB ansatz is Hamilton-Jacobi, made wave-mechanical by the prefactor $1/\sqrt{p}$. The bridge between classical mechanics and quantum mechanics is the $\hslash \to 0$ limit of WKB; conversely, Bohr-Sommerfeld is the only place in quantum mechanics where the classical action integral enters the spectrum directly.

• 05.02.04 Action-angle coordinates. EBK quantisation places one integer quantum number on each toroidal action of a Liouville-Arnold integrable system. The semiclassical bridge between integrable classical mechanics and quantum spectra is exactly the EBK formula.

• 12.04.02 Quantum harmonic oscillator. Bohr-Sommerfeld reproduces the exact oscillator spectrum $E_n=(n+1/2)\hslash \omega$, including the half-integer Maslov correction that yields the zero-point energy. The oscillator is the calibration example: any WKB tool that fails on the oscillator is broken.

• 12.06.01 Hydrogen atom. With the Langer correction $\ell (\ell +1)\to (\ell +1/2{)}^2$, Bohr-Sommerfeld reproduces the Bohr formula $E_n=-m{e}^4/(2{\hslash }^2{n}^2)$. Historically, hydrogen was the first nontrivial spectrum the old quantum theory (1913) could compute correctly; modern WKB recovers it with the correctness conditions made explicit.

• 12.07.01 Time-independent perturbation theory and 12.07.02 Fermi golden rule. WKB and perturbation theory are complementary approximate methods: perturbation theory expands in a coupling constant assumed small; WKB expands in $\hslash$ assumed small. The semi-classical limit is not the weak-coupling limit, and the two regimes capture different physics.

• Quantum tunnelling and Gamow theory. The exponential transmission formula $T\sim \exp (-2K/\hslash )$ underlies alpha decay (Gamow 1928), nuclear fission, electron field emission (Fowler-Nordheim 1928 Proc. Roy. Soc. A 119), scanning tunnelling microscopy, and Josephson tunnelling.

• Instantons and non-perturbative effects. The Euclidean version of WKB — Wick-rotated tunnelling — yields instantons, classical solutions of the imaginary-time equations of motion that mediate non-perturbative quantum corrections (Coleman 1977 Phys. Rev. D 15). Exact WKB is the precise mathematical home of the instanton/perturbation correspondence in 1D.

• Lagrangian Grassmannian and Maslov index. The half-integer offset in Bohr-Sommerfeld is the spectral manifestation of the topology of the Lagrangian Grassmannian $\Lambda (n)=U(n)/O(n)$, threaded into symplectic geometry, Floer theory, and the Atiyah-Singer index theorem.

Historical & philosophical context Master

The old quantum theory of 1913-1925 quantised classical orbits by imposing the Bohr-Sommerfeld condition $\oint pdq=nh$ on each independent action of an integrable system. Bohr 1913 applied it to the hydrogen atom, recovering the spectrum $E_n=-m{e}^4/(2{\hslash }^2{n}^2)$ that experimental work on the Lyman, Balmer, and Paschen lines had revealed (Balmer 1885 Ann. Phys. Chem. 25). Sommerfeld 1916 Ann. Phys. 51 extended the recipe to elliptical orbits and to relativistic momentum, producing the fine-structure formula that agreed with the experimental work of Paschen.

The old quantum theory worked for the harmonic oscillator and for the hydrogen atom but failed catastrophically for the helium atom — Bohr-Sommerfeld applied to the classical three-body problem gives the wrong ground-state energy. The reason, hidden inside the recipe, was that helium's classical dynamics is non-integrable, so EBK does not apply; the correct semi-classical treatment requires Gutzwiller's periodic-orbit theory developed half a century later. This failure was a major motivation for the wave-mechanical (Schrödinger) and matrix-mechanical (Heisenberg) reformulations of 1925-1926.

The WKB derivation of Bohr-Sommerfeld came almost immediately after Schrödinger's equation. Wentzel 1926 Zeits. f. Physik 38 ^{[source pending]}, Kramers 1926 Zeits. f. Physik 39 ^{[source pending]}, and Brillouin 1926 Comptes Rendus 183 ^{[source pending]} independently published the semiclassical expansion of the Schrödinger wavefunction in $\hslash$, recovering the Bohr-Sommerfeld condition with the explicit $1/2$ Maslov correction as a leading-order asymptotic result. Jeffreys 1925 Proc. London Math. Soc. 23 ^{[source pending]} had given the same asymptotic expansion two years earlier in a paper on linear second-order differential equations — a result the physicists were unaware of until later. Hence the name WKB (and, in mathematical-physics circles, WKBJ).

The connection-formula derivation as a matched asymptotic expansion was clarified by Langer 1937 Phys. Rev. 51 ^{[source pending]}, who introduced the rescaling $r\to e^s$ that makes the Airy matching uniform around radial turning points and corrects the radial Bohr-Sommerfeld spectrum for the centrifugal singularity. The Langer correction is what makes WKB reproduce the exact hydrogen spectrum; without it the formula gives the wrong principal quantum number.

Gamow 1928 Zeits. f. Physik 51 ^{[source pending]} applied the WKB tunnelling formula to alpha decay, deriving the Geiger-Nuttall rule $\log T_{1/2}\propto Z/\sqrt{E_{\alpha }}$ from a one-line semi-classical computation. The exponential sensitivity to barrier height explained why alpha-decay half-lives span more than 20 orders of magnitude across the periodic table. Gamow's 1928 paper is a model of theoretical physics: a single elementary formula, applied to one well-understood data set, settles an entire chapter of nuclear phenomenology.

Keller 1958 Ann. Phys. 4 ^{[source pending]} extended Bohr-Sommerfeld to $n$-dimensional integrable systems and identified the Maslov-index offset ${\mu }_i/4$ as the correct half-integer term per torus loop. Maslov 1965 reformulated the geometric setting in terms of Lagrangian submanifolds of the cotangent bundle, and the Maslov index became a central object of symplectic geometry. The modern home of these ideas is the microlocal-analysis programme of Hörmander, Duistermaat, Guillemin-Sternberg, and the symplectic Floer-theoretic work that followed.

Voros 1983 Ann. Inst. Henri Poincaré A 39 ^{[source pending]} launched the exact-WKB programme, identifying the resurgent / Borel-summable structure of the formal WKB series and showing that non-perturbative corrections of order $e^{-\text{action}/\hslash }$ are encoded in the Stokes phenomena. The exact-WKB framework, refined by Sato, Aoki, Kawai, Takei, and by Delabaere, Dillinger, Pham, has become a central tool in resurgent analysis and in the analysis of moduli spaces in supersymmetric gauge theory.

Landau and Lifshitz Volume 3 §§46-53 ^{[source pending]} gives the canonical textbook treatment of WKB and Bohr-Sommerfeld, with the connection formulas, the Maslov correction, the Gamow tunnelling factor, and the application to alpha decay all developed in roughly thirty pages. Berry and Mount 1972 Rep. Prog. Phys. 35 ^{[source pending]} wrote the canonical review of semi-classical approximations in wave mechanics, drawing the lines between WKB, EBK, and Gutzwiller and tying them to the geometry of caustics and Maslov indices.

Bibliography Master

Originator papers — WKB and connection formulas:

• Wentzel, G., "Eine Verallgemeinerung der Quantenbedingungen fur die Zwecke der Wellenmechanik", Zeitschrift fur Physik 38 (1926), 518-529.
• Kramers, H. A., "Wellenmechanik und halbzahlige Quantisierung", Zeitschrift fur Physik 39 (1926), 828-840.
• Brillouin, L., "La mecanique ondulatoire de Schrodinger; une methode generale de resolution par approximations successives", Comptes Rendus Acad. Sci. Paris 183 (1926), 24-26.
• Jeffreys, H., "On certain approximate solutions of linear differential equations of the second order", Proc. London Math. Soc. (2) 23 (1925), 428-436.
• Langer, R. E., "On the connection formulas and the solutions of the wave equation", Phys. Rev. 51 (1937), 669-676.

Old quantum theory and tunnelling:

• Bohr, N., "On the constitution of atoms and molecules", Phil. Mag. 26 (1913), 1-25.
• Sommerfeld, A., "Zur Quantentheorie der Spektrallinien", Ann. Phys. 51 (1916), 1-94.
• Gamow, G., "Zur Quantentheorie des Atomkernes", Zeitschrift fur Physik 51 (1928), 204-212.
• Fowler, R. H. & Nordheim, L., "Electron emission in intense electric fields", Proc. Roy. Soc. A 119 (1928), 173-181.

EBK quantisation, Maslov index, exact WKB:

• Keller, J. B., "Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems", Annals of Physics 4 (1958), 180-188.
• Maslov, V. P., Theory of Perturbations and Asymptotic Methods (Izdat. Moskov. Gos. Univ., Moscow, 1965; French transl. Dunod, 1972).
• Gutzwiller, M. C., "Periodic orbits and classical quantization conditions", J. Math. Phys. 12 (1971), 343-358.
• Voros, A., "The return of the quartic oscillator: the complex WKB method", Annales de l'Institut Henri Poincare A 39 (1983), 211-338.
• Bender, C. M. & Wu, T. T., "Anharmonic oscillator", Phys. Rev. 184 (1969), 1231-1260.
• Delabaere, E., Dillinger, H. & Pham, F., "Exact semiclassical expansions for one-dimensional quantum oscillators", J. Math. Phys. 38 (1997), 6126-6184.
• Berry, M. V. & Mount, K. E., "Semiclassical approximations in wave mechanics", Reports on Progress in Physics 35 (1972), 315-397.

Textbooks:

• Landau, L. D. & Lifshitz, E. M., Quantum Mechanics: Non-Relativistic Theory, 3rd ed. (Pergamon, 1977), Ch. VII (§§46-53). The canonical physicist's treatment of WKB and Bohr-Sommerfeld in roughly thirty pages.
• Griffiths, D. J. & Schroeter, D. F., Introduction to Quantum Mechanics, 3rd ed. (Cambridge, 2018), Ch. 9. Pedagogical introduction at undergraduate level.
• Bender, C. M. & Orszag, S. A., Advanced Mathematical Methods for Scientists and Engineers (Springer, 1999), Ch. 10. The asymptotic-analysis perspective on WKB and matched expansions.
• Messiah, A., Quantum Mechanics, Vols. I-II (North-Holland, 1961; Dover reprint 1999), Ch. VI. Semiclassical methods at graduate-text depth.
• Galindo, A. & Pascual, P., Quantum Mechanics II (Springer, 1991), Ch. 9. Modern graduate treatment with emphasis on rigour.
• Olver, F. W. J., Asymptotics and Special Functions (Academic Press, 1974). Standard reference for the Airy-function asymptotics underpinning the connection formulas.

Modern reviews and mathematical physics:

• Robert, D., Autour de l'Approximation Semi-Classique (Birkhauser, 1987). Semiclassical analysis from the microlocal perspective.
• Dimassi, M. & Sjostrand, J., Spectral Asymptotics in the Semi-Classical Limit, LMS Lecture Note Series 268 (Cambridge, 1999). The contemporary mathematical treatment.
• Arnold, V. I., Mathematical Methods of Classical Mechanics, 2nd ed. (Springer, 1989), Appendix 11. EBK and the Maslov index in the symplectic-geometry framework.
• Zworski, M., Semiclassical Analysis, GSM 138 (AMS, 2012). The modern graduate textbook synthesising microlocal and semiclassical analysis.