---
title: The WKB Approximation
module: Approximation Methods for Bound States
moduleNumber: 10
lessonNumber: 5
order: 1005
summary: >
  When the potential varies slowly on the scale of the de Broglie wavelength,
  the wavefunction is locally a plane wave with a position-dependent wavelength.
  This semiclassical picture builds the wavefunction from the classical
  momentum, breaks down at the turning points where the momentum vanishes, and
  is repaired there by connection formulas. The result recovers the
  Bohr–Sommerfeld quantization rule with its half-integer correction and gives
  the exponential tunneling rate through a smooth barrier, the Gamow factor.
topics: [Approximation Methods for Bound States]
sources:
  - book: Griffiths & Schroeter
    ref: "Ch. 9 — The WKB Approximation; §9.1 The Classical Region, §9.2 Tunneling, §9.3 Connection Formulas"
  - book: Shankar
    ref: "Ch. 16; §16.2 The WKB Method"
  - book: Sakurai & Napolitano
    ref: "Ch. 5; §5.6 Approximation Methods for Bound States: the WKB Approximation"
draft: false
---

Perturbation theory and the variational method both start from a solvable
problem. The WKB approximation, named for Wentzel, Kramers, and Brillouin,
starts instead from the classical limit. When the potential changes slowly
enough that the de Broglie wavelength is nearly constant over one wavelength, the
wavefunction is locally sinusoidal with a wavelength set by the local classical
momentum. The method builds the full wavefunction by stitching these local
plane waves together, fails only at the turning points where the classical
momentum vanishes, and is patched there by matching to an exact local solution.
It yields the quantization condition of the
[old quantum theory](/quantum-mechanics/old-quantum-theory/the-old-quantum-theory-bohr-and-sommerfeld)
with the correct half-integer offset, and the tunneling rate through a barrier of
arbitrary shape.

## The semiclassical wavefunction

In a region where $E > V(x)$ the classical momentum
$p(x) = \sqrt{2m[E - V(x)]}$ is real. Write the stationary wavefunction in
amplitude–phase form $\psi(x) = A(x)\,e^{i\phi(x)}$ and substitute into the
time-independent Schrödinger equation $\psi'' = -(p^2/\hbar^2)\psi$. Separating
real and imaginary parts and dropping the term with $A''$ (small when the
amplitude varies slowly) gives two equations: the phase obeys
$\phi'(x) = \pm p(x)/\hbar$, and the amplitude obeys $(A^2\phi')' = 0$, so
$A \propto 1/\sqrt{p}$.

> **Definition (WKB wavefunction).** In a classically allowed region the
> semiclassical wavefunction is
> $$\psi(x) \approx \frac{C}{\sqrt{p(x)}}\,\exp\!\left(\pm\frac{i}{\hbar}\int^x p(x')\,\d x'\right),$$
> a plane wave whose local wavelength $\lambda(x) = h/p(x)$ and amplitude both
> track the classical momentum.

The amplitude law has a transparent classical reading. The probability density
is $\lvert\psi\rvert^2 \propto 1/p \propto 1/v$: the particle is most likely to be
found where it moves slowest, exactly as a classical particle spends the most
time in the slow parts of its orbit. The quantum wavefunction inherits the
classical dwelling time.

$$
% caption: In the classically allowed region the WKB wavefunction oscillates
% with a local wavelength set by the momentum and an amplitude that swells toward
% the turning points where the particle slows and the classical dwelling time
% grows.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% potential well
\draw[black, very thick] plot[domain=-2.5:2.5, samples=80] (\x, {0.45*\x*\x});
\node[anchor=west, black!70] at (2.0,2.5) {$V(x)$};
% energy line
\draw[black, thick] (-2.2,2.2) -- (2.2,2.2);
\node[anchor=east, black!70] at (-2.2,2.2) {$E$};
% turning points
\fill[black] (-2.21,2.2) circle (1.6pt);
\fill[black] (2.21,2.2) circle (1.6pt);
% oscillatory wavefunction (amplitude grows toward edges)
\draw[acc, very thick] plot[domain=-2.05:2.05, samples=240]
  (\x, {0.7 + (0.20 + 0.11*\x*\x)*sin(500*\x - 30*\x*\x*\x)});
\node[anchor=north, acc] at (0,0.30) {WKB wave};
\end{tikzpicture}
$$

## The validity condition

The neglected term $A''$ is small precisely when the wavelength changes little
over one wavelength. Quantifying that gives the domain of the method.

> **Definition (WKB validity).** The approximation holds where the fractional
> change of the wavelength over one wavelength is small,
> $$
> \left\lvert\frac{\d\lambda}{\d x}\right\rvert \ll 1,
> \qquad\text{equivalently}\qquad
> \frac{m\hbar\,\lvert V'(x)\rvert}{p(x)^3} \ll 1.
> $$

The condition fails wherever $p \to 0$, that is at the classical **turning
points** where $E = V(x)$. There the wavelength diverges, the amplitude
$1/\sqrt{p}$ blows up, and the local-plane-wave picture is meaningless. Every WKB
calculation is organized around these points: the wavefunction is built from
WKB pieces in the regions between them and repaired across them.

## The classically forbidden region

Where $E < V(x)$ the momentum is imaginary. Writing
$\lvert p(x)\rvert = \sqrt{2m[V(x) - E]}$, the same derivation gives real
exponentials instead of oscillations,

$$
\psi(x) \approx \frac{C}{\sqrt{\lvert p(x)\rvert}}\,
\exp\!\left(\pm\frac{1}{\hbar}\int^x \lvert p(x')\rvert\,\d x'\right).
$$

The decaying solution is the tunneling tail: a wavefunction penetrating a
barrier is attenuated by the exponential of the accumulated $\lvert p\rvert$. The
growing solution is discarded for a barrier of finite width only after matching,
since it would dominate; inside a wide barrier the decaying piece carries the
transmitted amplitude.

## Connection formulas

At a turning point the WKB forms on the two sides — oscillatory where allowed,
exponential where forbidden — must be joined. Near an isolated turning point
$x = a$ the potential is nearly linear, $V(x) \approx E + V'(a)(x - a)$, and the
Schrödinger equation reduces to the **Airy equation**, whose solution is known
exactly. Matching the Airy function's asymptotic forms to the WKB expressions on
each side fixes the relative amplitudes and, decisively, a phase.

> **Theorem (Connection formula).** At a turning point $x = a$ with the forbidden
> region on the left and the allowed region on the right, the decaying
> exponential connects to a cosine with a $\pi/4$ phase lag,
> $$
> \frac{1}{\sqrt{\lvert p\rvert}}\exp\!\left(-\frac{1}{\hbar}\int_x^a\lvert p\rvert\,\d x'\right)
> \;\longleftrightarrow\;
> \frac{2}{\sqrt{p}}\cos\!\left(\frac{1}{\hbar}\int_a^x p\,\d x' - \frac{\pi}{4}\right).
> $$
> A mirror-image formula holds at a turning point with the allowed region on the
> left.

The $\pi/4$ phase shift at each turning point is the content of the connection
formula and the origin of the half-integer in the quantization rule below. It
records the smooth handoff, through the Airy region, between an oscillation and
an exponential.

$$
% caption: Across a turning point the linearized potential gives an exact Airy
% solution that interpolates the oscillatory allowed-side WKB wave and the
% decaying forbidden-side exponential, contributing a fixed phase lag of a
% quarter cycle.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% vertical line at turning point
\draw[black, dashed] (0,-1.4) -- (0,1.6);
\node[anchor=south, black!70] at (0,1.6) {turning point};
% allowed side oscillation (right)
\draw[acc, very thick] plot[domain=0:3.2, samples=120] (\x, {0.9*sin(360*\x/1.4 - 45)});
\node[anchor=north, acc] at (2.2,-0.9) {oscillatory (allowed)};
% forbidden side decay (left)
\draw[black, very thick, dashed] plot[domain=-3.0:0, samples=80] (\x, {0.9*exp(1.1*\x)});
\node[anchor=south, black!70] at (-2.0,0.7) {exponential (forbidden)};
\end{tikzpicture}
$$

## The quantization condition

For a particle bound in a well with two turning points $a < b$, the allowed
region lies between them and the forbidden regions on either side demand decaying
solutions. Applying the connection formula at each turning point yields a cosine
referenced to $a$ and another referenced to $b$; consistency of the single
wavefunction in the middle forces the total accumulated phase to be a multiple of
$\pi$.

> **Theorem (Bohr–Sommerfeld quantization).** A smooth potential well with two
> turning points has bound states where the enclosed action satisfies
> $$
> \int_a^b p(x)\,\d x = \left(n + \tfrac12\right)\pi\hbar,
> \qquad n = 0, 1, 2, \dots,
> $$
> equivalently, over a full classical period,
> $$\oint p\,\d x = \left(n + \tfrac12\right)h.$$

The rule is the old quantum theory's $\oint p\,\d x = nh$ corrected by the
half-integer. Each of the two soft turning points contributes a $\pi/4$ phase,
together shifting $n$ to $n + \tfrac12$. The integer $n$ counts the nodes of the
wavefunction, so the quantization is a statement that a whole number of
half-wavelengths, plus the two quarter-wave turning-point corrections, fits
between the walls.

$$
% caption: The quantization condition is the statement that the classical orbit
% encloses a half-integer number of units of Planck's constant in phase space;
% each closed loop in the x-p plane is an allowed state.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black, ->] (-3.2,0) -- (3.2,0) node[right, black!70] {$x$};
\draw[black, ->] (0,-2.2) -- (0,2.2) node[above, black!70] {$p$};
% phase-space ellipse
\draw[acc, very thick] (0,0) ellipse (2.6 and 1.6);
\fill[acc!10] (0,0) ellipse (2.6 and 1.6);
\draw[acc, very thick] (0,0) ellipse (2.6 and 1.6);
\node[acc] at (0,0) {area $= (n + \tfrac12)h$};
% turning points
\fill[black] (-2.6,0) circle (1.8pt);
\fill[black] (2.6,0) circle (1.8pt);
\node[anchor=north east, black!70] at (-2.6,0) {$a$};
\node[anchor=north west, black!70] at (2.6,0) {$b$};
\end{tikzpicture}
$$

> **Worked example.** For the harmonic oscillator $V(x) = \tfrac12 m\omega^2 x^2$
> the turning points are $\pm\sqrt{2E/m\omega^2}$ and the action integral is
> $$
> \int_a^b p\,\d x = \int_a^b\sqrt{2m\left(E - \tfrac12 m\omega^2 x^2\right)}\,\d x
> = \frac{\pi E}{\omega}.
> $$
> Setting it equal to $(n + \tfrac12)\pi\hbar$ gives
> $$E_n = \left(n + \tfrac12\right)\hbar\omega,$$
> the exact spectrum. WKB is exact for the oscillator because the half-integer it
> supplies is precisely the zero-point offset, and the semiclassical action
> happens to be evaluated without error for a quadratic potential.

## Tunneling and the Gamow factor

For a barrier where $V(x) > E$ between turning points $a$ and $b$, the
wavefunction decays across the classically forbidden region. The transmitted
amplitude is suppressed by the exponential of the accumulated $\lvert p\rvert$,
and the transmission probability is its square.

> **Theorem (WKB tunneling).** Through a smooth barrier the transmission
> probability is
> $$
> T \approx e^{-2\gamma}, \qquad
> \gamma = \frac{1}{\hbar}\int_a^b \lvert p(x)\rvert\,\d x
> = \frac{1}{\hbar}\int_a^b\sqrt{2m[V(x) - E]}\,\d x,
> $$
> valid when $\gamma \gg 1$ (a thick, high barrier).

The exponent $\gamma$ is the area under the forbidden-region momentum, so a wider
or taller barrier suppresses tunneling exponentially. This is the semiclassical
generalization of the rectangular-barrier result of the
[tunneling lesson](/quantum-mechanics/wave-mechanics-1d/barrier-penetration-and-quantum-tunneling)
to a barrier of any shape.

$$
% caption: WKB tunneling accumulates the imaginary momentum across the forbidden
% region; the transmission is the exponential of twice the shaded area under the
% square-root of V minus E between the two turning points.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% barrier
\draw[black, very thick] plot[domain=-2.6:2.6, samples=100] (\x, {2.4*exp(-0.6*\x*\x)});
\node[anchor=west, black!70] at (2.0,1.3) {$V(x)$};
% energy line
\draw[black, thick] (-2.8,1.2) -- (2.8,1.2);
\node[anchor=east, black!70] at (-2.8,1.2) {$E$};
% shaded forbidden region under barrier above E
\fill[acc!14] plot[domain=-1.28:1.28, samples=60] (\x, {2.4*exp(-0.6*\x*\x)}) -- (1.28,1.2) -- (-1.28,1.2) -- cycle;
\node[acc] at (0,1.75) {area};
% turning points
\fill[black] (-1.28,1.2) circle (1.8pt);
\fill[black] (1.28,1.2) circle (1.8pt);
\node[anchor=north, black!70] at (-1.28,1.15) {$a$};
\node[anchor=north, black!70] at (1.28,1.15) {$b$};
\end{tikzpicture}
$$

> **Worked example.** In alpha decay the alpha particle tunnels out through the
> Coulomb barrier of the daughter nucleus. With $V(r) = 2Ze^2/4\pi\epsilon_0 r$
> (daughter charge $Ze$), the forbidden region runs from the nuclear radius $r_1$
> to the outer turning point $r_2 = 2Ze^2/4\pi\epsilon_0 E$ where $V = E$, and the
> Gamow exponent is
> $$
> \gamma = \frac{1}{\hbar}\int_{r_1}^{r_2}\sqrt{2m\left(\frac{2Ze^2}{4\pi\epsilon_0 r} - E\right)}\,\d r
> \;\approx\; \frac{\pi Z e^2}{4\pi\epsilon_0\hbar}\sqrt{\frac{2m}{E}}
> $$
> in the limit $r_1 \ll r_2$. The decay constant is $\lambda \sim f\,e^{-2\gamma}$,
> with $f$ the frequency at which the alpha strikes the barrier. Because
> $\gamma \propto Z/\sqrt{E}$, a modest increase in the decay energy $E$ shortens
> the half-life by many orders of magnitude. This exponential sensitivity is the
> Geiger–Nuttall law, $\log_{10}\tau_{1/2} \propto Z/\sqrt{E}$, which a
> rectangular barrier of fixed width cannot reproduce.

The Gamow factor governs processes from alpha decay, where the alpha particle
tunnels through the Coulomb barrier of the daughter nucleus, to field emission
and the ammonia inversion. In alpha decay the exponential dependence of $\gamma$
on energy converts a modest spread in decay energies into the enormous range of
observed half-lives, the Geiger–Nuttall law, which the rectangular-barrier model
cannot reproduce.

The three methods of this module cover the ground where exact solutions run out.
[Perturbation theory](/quantum-mechanics/approximation-methods/time-independent-perturbation-theory)
handles a small correction to a solved problem, the
[variational method](/quantum-mechanics/approximation-methods/the-variational-method)
bounds a ground state with no small parameter, and WKB works wherever the
potential is smooth on the quantum scale. Together they turn the exactly solvable
core of quantum mechanics into a toolkit for real atoms, molecules, and nuclei.

[^griffiths-wkb]: **Griffiths & Schroeter**, _Introduction to Quantum Mechanics_ 3rd ed., Ch. 9 — the semiclassical wavefunction and validity condition (§9.1), tunneling and the Gamow factor (§9.2), the connection formulas and Bohr–Sommerfeld quantization with the half-integer correction (§9.3). Cambridge, 2018.
[^shankar-wkb]: **Shankar**, _Principles of Quantum Mechanics_ 2nd ed., §16.2 — the WKB series, the turning-point problem solved through the Airy equation, and the quantization rule. Springer, 1994.
[^sakurai-wkb]: **Sakurai & Napolitano**, _Modern Quantum Mechanics_ 3rd ed., §5.6 — the WKB approximation for bound states and the connection to the classical action. Cambridge, 2021.