WKB approximation: tunneling, Bohr-Sommerfeld quantization, and connection formulas

Intuition Beginner

Most quantum problems cannot be solved exactly. The particle in a box, the harmonic oscillator, the hydrogen atom — these rare exceptions have closed-form wavefunctions and energy levels. Nearly any other potential shape forces you to reach for approximation methods.

The WKB method (named for Wentzel, Kramers, and Brillouin) is the standard approximation when the potential $V(x)$ changes slowly compared to the particle's local de Broglie wavelength. A particle with energy $E$ in a region where $V(x)<E$ moves with local momentum $p(x)=\sqrt{2m(E-V(x))}$ and local wavelength $\lambda (x)=2\pi \hslash /p(x)$.

If $V(x)$ barely changes over one wavelength, the particle behaves almost like a free particle whose momentum drifts slowly. Its wavefunction oscillates locally like $e^{ip(x)\cdot x/\hslash }$ with an amplitude that adjusts to conserve probability.

The method fails at classical turning points, the positions $x=a$ where $E=V(a)$. There the local momentum $p(a)=0$, the wavelength diverges, and the slow-variation condition collapses. A separate treatment — matching the oscillatory WKB wavefunction to the exact Airy-function solution of a linearized potential near the turning point — bridges the allowed and forbidden regions. These are the connection formulas.

For a particle trapped between two turning points, requiring the wavefunction to match at both endpoints produces the Bohr-Sommerfeld quantization condition: the action integral $\oint pdq$ over one classical period equals $(n+1/2)h$. This formula, the workhorse of the old quantum theory (1913-1925), is recovered and made rigorous by WKB.

For a barrier where $E<V$, the WKB wavefunction penetrates the classically forbidden region with an amplitude that decays exponentially. The resulting tunneling probability $T\approx e^{-2K/\hslash }$, where $K$ is the integrated action across the barrier, explains alpha decay, scanning tunneling microscopy, and many other phenomena.

Visual Beginner

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

Worked example Beginner

Bohr-Sommerfeld for the harmonic oscillator. The oscillator has $V(x)=\frac{1}{2}m{\omega }^2{x}^2$. At energy $E$, the turning points are $x=\pm \sqrt{2E/(m{\omega }^2)}$. The phase-space orbit satisfies $p^2/(2m)+\frac{1}{2}m{\omega }^2{x}^2=E$, which is an ellipse with semi-axes $A=\sqrt{2mE}$ (momentum) and $B=\sqrt{2E/(m{\omega }^2)}$ (position).

The area enclosed by this ellipse is the action integral:

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

The Bohr-Sommerfeld condition $\oint pdq=2\pi \hslash (n+\frac{1}{2})$ gives:

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

This is the exact harmonic-oscillator spectrum. The half-integer offset is essential: without it the ground state would sit at $E_0=0$, contradicting the zero-point energy $\frac{1}{2}\hslash \omega$.

Check your understanding Beginner

Formal definition Intermediate+

Consider the one-dimensional time-independent Schrodinger equation

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

Write the wavefunction as $\psi (x)=A(x)\exp (iS(x)/\hslash )$ with real $S$ and $A$. Substituting and separating real and imaginary parts:

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

The second equation is probability current conservation: $j=A^2{S}^{\prime }/m=\text{const}$.

Expand $S$ as a formal series $S=S_0+\hslash S_1+{\hslash }^2{S}_2+\cdots$.

Leading order (${\hslash }^0$): $(S_0^{\prime }{)}^2=2m(E-V(x))$, so $S_0=\pm {\int }^{x}p(y)dy$ where $p(x)=\sqrt{2m(E-V(x))}$ is the classical local momentum.

Next-to-leading order (${\hslash }^1$): the current conservation equation with $S_0^{\prime }=\pm p$ gives $A=C/\sqrt{p(x)}$.

WKB wavefunction (allowed region, $E>V$):

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

WKB wavefunction (forbidden region, $E<V$): set $|p(x)|=\sqrt{2m(V(x)-E)}$:

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

Validity criterion: $\hslash |dp/dx|\ll p^2$, equivalently $|d{\lambda }_{\text{dB}}/dx|\ll 1$. This fails at turning points where $p\to 0$.

Connection formulas

Near a turning point $x=a$ with $V(a)=E$, $V^{\prime }(a)>0$, expand $V(x)-E\approx V^{\prime }(a)(x-a)$. The Schrodinger equation becomes the Airy equation in the rescaled variable $z=(x-a)/\ell$ with $\ell =({\hslash }^2/(2m{V}^{\prime }(a)){)}^{1/3}$. The decaying solution $\text{Ai}(z)$ has asymptotics:

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

Identifying $\frac{2}{3}|z{|}^{3/2}={\int }_x^{a}pdy/\hslash$ (allowed side) and $\frac{2}{3}{z}^{3/2}={\int }_a^x|p|dy/\hslash$ (forbidden side), the Langer connection formula reads:

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

The arrow is one-directional: from the small decaying exponential on the forbidden side to a definite cosine on the allowed side. Reversing it loses information because a tiny admixture of the growing exponential, invisible on the forbidden side, dominates after a short distance.

Counterexamples to common slips

• The WKB approximation is not an expansion in the potential strength. It is an expansion in $\hslash$, valid when the de Broglie wavelength is short compared to the scale of potential variation. A weak but rapidly varying potential can violate WKB; a strong but slowly varying one can satisfy it.
• The connection formula is one-directional. Applying it from the allowed side to the forbidden side (choosing which decaying exponential to keep) requires additional information about the boundary conditions. Starting from the forbidden side and connecting to the allowed side is the safe direction.
• The Bohr-Sommerfeld condition is not exact for generic potentials. It is a leading-order semiclassical result. The harmonic oscillator and the Morse potential are exceptions where the leading WKB answer happens to be exact.

Key theorem with proof Intermediate+

Theorem (Bohr-Sommerfeld quantization). Let $V(x)$ be a smooth potential with a single well bounded by two turning points $a<b$ where $V(a)=V(b)=E$, with $V^{\prime }(a)<0$ and $V^{\prime }(b)>0$. The WKB bound-state energies satisfy

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

Proof. Apply the connection formula at each turning point and demand consistency.

Left turning point $a$. For $x<a$, the bound-state wavefunction must decay:

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

The connection formula (with allowed region to the right of $a$) gives:

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

Right turning point $b$. By the same reasoning (forbidden to the right):

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

Consistency. Using ${\int }_x^{b}pdy={\int }_a^{b}pdy-{\int }_a^{x}pdy$ in $(\star \star )$:

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

Setting this equal to $(\star )$ and using $\cos \alpha =\cos \beta$ iff $\alpha =\pm \beta +k\pi$, the $x$-dependent terms match and the constant piece gives:

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

This is ${\int }_a^{b}pdx=\pi \hslash (n+1/2)$. The two $-\pi /4$ shifts sum to $-\pi /2$, producing the Maslov half-integer. $\square$

Bridge. The Bohr-Sommerfeld condition builds toward 12.07.04 where EBK quantization extends the recipe to multi-dimensional integrable systems via the Maslov index on the Lagrangian Grassmannian, and appears again in 12.04.02 where the harmonic oscillator provides the calibration test that any semiclassical method must pass.

Exercises Intermediate+

Advanced results Master

Higher-order WKB and exact WKB

The leading 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. The series is asymptotic, not convergent: the truncated $N$-th partial sum approximates the true wavefunction with error $O({\hslash }^{N+1})$ for small enough $\hslash$.

Exact WKB, developed by Voros, Sato-Aoki-Kawai, and Delabaere-Dillinger-Pham, addresses the divergence through resurgent analysis. The formal series carries non-perturbative information in its Stokes phenomena: where partial sums cross specific lines in the complex $\hslash$-plane, exponentially small corrections of order $\exp (-\text{action}/\hslash )$ must be added. The full exact-WKB expansion reconstructs the exact spectrum order by order through Borel-Laplace resummation and alien calculus. The machinery has applications to anharmonic oscillators (Bender-Wu 1969), to instanton counting in gauge theories (Dunne-Unsal 2014), and to the quantum monodromy of integrable systems.

The Gutzwiller trace formula

For non-integrable systems the Bohr-Sommerfeld recipe breaks down: phase-space tori dissolve under chaos and there are no canonical action variables to quantize. The Gutzwiller trace formula (1971) replaces Bohr-Sommerfeld for chaotic systems. It expresses the semiclassical density of states as a sum over classical periodic orbits:

$$\rho (E)\sim \bar{\rho }(E)+\sum _{\text{pp}}{A}_{\text{pp}}(E)\cos \left(\frac{S_{\text{pp}}(E)}{\hslash }-\frac{\pi {\mu }_{\text{pp}}}{2}\right),$$

where $S_{\text{pp}}$ is the classical action along the periodic orbit, ${\mu }_{\text{pp}}$ is its Maslov index, and $A_{\text{pp}}$ is a stability amplitude. The formula connects quantum spectral statistics to classical periodic orbits and underlies the connection between quantum chaos and random-matrix theory (Bohigas-Giannoni-Schmit conjecture 1984).

Uniform asymptotic approximations

The Langer connection formula is a local approximation near a single turning point. When two turning points approach each other (e.g., near the top of a barrier or at the dissociation threshold of a well), the Airy matching becomes inaccurate and a uniform approximation is needed. The standard technique replaces the Airy function by a parabolic cylinder function (for a well that is almost open) or a Weber function (for a barrier that is nearly transparent). The uniform approximations, developed by Berry, Connor, and others, remain valid through the transition region where separate connection formulas fail.

WKB for the radial Schrodinger equation

The radial equation has an effective potential $V_{\text{eff}}(r)=V(r)+{\hslash }^2\ell (\ell +1)/(2m{r}^2)$. The centrifugal singularity at $r=0$ requires the Langer correction: substituting $\ell (\ell +1)\to (\ell +1/2{)}^2$. The correction follows from the change of variable $r=e^s$ which transforms the centrifugal term into a smooth potential near $s\to -\infty$, producing an effective angular-momentum quantum number $\ell +1/2$ in the WKB phase integrals. Without the Langer correction, Bohr-Sommerfeld gives the wrong hydrogen spectrum.

Connections Master

• 12.04.02 Harmonic oscillator. Bohr-Sommerfeld reproduces the exact spectrum $E_n=(n+1/2)\hslash \omega$ including the zero-point energy. The oscillator is the calibration example: the quadratic potential is locally identical to its linear approximation at every point, so WKB is exact.

• 12.04.01 Particle in a box. The infinite square well tests WKB with hard-wall (Dirichlet) boundary conditions. The Maslov offset changes from $1/2$ to $1$ for two hard walls, and the Bohr-Sommerfeld result is again exact.

• 12.07.04 WKB and Bohr-Sommerfeld (perturbation chapter). Extends this unit's foundations to EBK quantization in multiple dimensions, the Maslov index as a topological invariant, exact WKB via resurgence, and the Gutzwiller trace formula for chaotic systems.

• 12.06.01 Hydrogen atom. With the Langer correction $\ell (\ell +1)\to (\ell +1/2{)}^2$, Bohr-Sommerfeld recovers the exact Bohr formula $E_n=-m{e}^4/(2{\hslash }^2{n}^2)$.

• 12.01.01 Wave-particle duality. The de Broglie wavelength $\lambda =h/p$ is the physical input for WKB: the approximation treats the particle as a wave whose local wavelength is set by the classical momentum. WKB is the quantitative framework that makes the wave-particle duality precise for bound states and tunneling.

• Tunneling phenomena. The exponential transmission $T\sim e^{-2K/\hslash }$ underlies alpha decay (Gamow 1928), scanning tunneling microscopy, Josephson junctions, field emission (Fowler-Nordheim 1928), and nuclear fission. The WKB tunneling formula is one of the most widely used results in quantum physics.

Historical notes Master

The old quantum theory (1913-1925) imposed the quantization condition $\oint pdq=nh$ on classical orbits. Bohr 1913 derived the hydrogen spectrum; Sommerfeld 1916 extended the recipe to relativistic elliptical orbits, producing the fine-structure formula. The theory failed for helium (non-integrable three-body problem) and had no systematic procedure for transition rates.

Wentzel, Kramers, and Brillouin (all 1926) independently derived the semiclassical expansion of the Schrodinger wavefunction, recovering Bohr-Sommerfeld with the explicit $+1/2$ Maslov correction as a leading-order result. Jeffreys (1925) had obtained the same asymptotic expansion two years earlier for linear second-order ODEs, unrecognized by the physics community until later — hence the name WKBJ in mathematical-physics circles.

Langer (1937) clarified the connection-formula derivation as a matched asymptotic expansion, introducing the substitution $r=e^s$ that makes the Airy matching uniform near radial turning points and produces the Langer correction for the centrifugal singularity.

Gamow (1928) applied the WKB tunneling formula to alpha decay, deriving the Geiger-Nuttall relation from a single computation. The exponential sensitivity to barrier height explained the 20+ orders-of-magnitude spread in alpha-decay half-lives across the periodic table — a triumph of theoretical physics.

Keller (1958) extended Bohr-Sommerfeld to $n$-dimensional integrable systems with the Maslov-index offset. Maslov (1965) reformulated the geometric setting in terms of Lagrangian submanifolds. Voros (1983) launched the exact-WKB program through resurgent analysis. Gutzwiller (1971) developed the periodic-orbit trace formula for non-integrable (chaotic) systems, the successor to Bohr-Sommerfeld when classical tori no longer exist.

Bibliography Master

Originator papers:

• Wentzel, G., "Eine Verallgemeinerung der Quantenbedingungen fur die Zwecke der Wellenmechanik", Z. Phys. 38 (1926), 518-529.
• Kramers, H. A., "Wellenmechanik und halbzahlige Quantisierung", Z. Phys. 39 (1926), 828-840.
• Brillouin, L., "La mecanique ondulatoire de Schrodinger", C. R. 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.
• Gamow, G., "Zur Quantentheorie des Atomkernes", Z. Phys. 51 (1928), 204-212.

Textbooks:

• Griffiths, D. J. & Schroeter, D. F., Introduction to Quantum Mechanics, 3rd ed. (Cambridge, 2018), Ch. 8.
• Sakurai, J. J. & Napolitano, J., Modern Quantum Mechanics, 3rd ed. (Cambridge, 2021), §2.5.
• Messiah, A., Quantum Mechanics, Vol. 1 (North-Holland, 1961; Dover, 1999), Ch. VI.
• Bender, C. M. & Orszag, S. A., Advanced Mathematical Methods for Scientists and Engineers (Springer, 1999), Ch. 10.
• Landau, L. D. & Lifshitz, E. M., Quantum Mechanics: Non-Relativistic Theory, 3rd ed. (Pergamon, 1977), §§46-53.

Modern references:

• Berry, M. V. & Mount, K. E., "Semiclassical approximations in wave mechanics", Rep. Prog. Phys. 35 (1972), 315-397.
• Keller, J. B., "Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems", Ann. Phys. 4 (1958), 180-188.
• 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", Ann. Inst. H. Poincare A 39 (1983), 211-338.
• Zworski, M., Semiclassical Analysis, GSM 138 (AMS, 2012).