WKB and Averaging for Slowly Varying Systems

A great many physical systems are built from a fast oscillation carried on a slow drift: a pendulum whose string is reeled in over many swings, a charged particle spiralling around a field line while its guiding centre wanders, a quantum wavefunction whose wavelength stretches and shrinks as the potential changes underfoot. In every case a single small parameter $\varepsilon$ measures the ratio of the fast period to the slow time scale. Feed such a problem to naive perturbation theory and it betrays you: the expansion sprouts secular terms — pieces that grow like $\varepsilon t$ and swamp the answer once $t\sim 1/\varepsilon$ , exactly the time scale one cares about. The art of this chapter is to resum those secular terms before they are generated. The averaging method, crystallised by Krylov, Bogoliubov and Mitropolsky, does this by a change of viewpoint. Write the motion in amplitude–phase or action–angle variables; notice that the angle races while the amplitude and action merely crawl; then replace the fast angle everywhere by its cycle average. The leftover averaged equation is autonomous, easy to integrate, and — a theorem guarantees — uniformly accurate to $O(\varepsilon)$ across the whole slow time. When the perturbation is a slow drift of the oscillator's own parameters the same average delivers the classical adiabatic theorem: the action $J=\oint p\,\mathrm dq$ is conserved to every algebraic order in $\varepsilon$ . And the spatial WKB problem $\varepsilon^2y''+Qy=0$ is averaging in disguise — the eikonal phase and the $Q^{-1/4}$ amplitude are precisely the conservation of a local action density.

Key takeaways

Averaging replaces a weakly perturbed oscillator $\dot x=\varepsilon f(x,\theta)$ by the cycle-averaged drift $\dot{\bar x}=\varepsilon\bar f(\bar x)$ , killing secular terms and staying accurate to $O(\varepsilon)$ over $t\sim1/\varepsilon$ ; a near-identity change of variables makes this rigorous, forcing $\bar f$ to be the mean.
When the oscillator's parameters drift slowly, the action $J=\oint p\,\mathrm dq$ is an adiabatic invariant — conserved to all algebraic orders, with only an exponentially small $\sim e^{-c/\varepsilon}$ leak.
The spatial WKB solution $Q^{-1/4}\exp(\tfrac{i}{\varepsilon}\!\int\!\sqrt Q)$ is the same idea in space: the $Q^{-1/4}$ prefactor conserves a local current, the spatial twin of $E/\omega$ .
Averaging collapses near resonances and at turning points, where the fast and slow scales merge and one needs resonance analysis or connection formulae instead.
The action invariant underpins Bohr–Sommerfeld quantization, plasma confinement, and adiabatic state transfer.
Key idea: separate fast from slow — work in phase–amplitude or action–angle variables, recognise that the amplitude and action drift while the phase races, and replace the fast phase by its cycle average; the averaged equation captures the true slow evolution and eliminates the spurious secular growth of naive perturbation theory.

The core insight

Separate fast from slow: work in phase–amplitude or action–angle variables, recognise that the amplitude and action drift slowly while the phase races, and replace the fast phase by its cycle average — the averaged equation captures the genuine slow evolution and eliminates the secular terms of naive perturbation theory.

Where this lives today

The action invariant is quietly at work across modern physics. In magnetic-confinement fusion and in the Earth's radiation belts, the magnetic moment $\mu=mv_\perp^2/2B$ — the action of a charged particle's fast Larmor gyration — is an adiabatic invariant that keeps plasma trapped: as a particle spirals into a stronger field its perpendicular energy must rise to hold $\mu$ fixed, and it is reflected, the principle behind the magnetic mirror and the trapping of MeV electrons in the Van Allen belts. In atomic and quantum-information physics, slow parameter sweeps carry a system adiabatically along an instantaneous eigenstate: STIRAP (stimulated Raman adiabatic passage) transfers population between atomic levels with near-unit fidelity, and the exponentially small $\delta J\sim e^{-c/\varepsilon}$ is exactly the Landau–Zener leakage that adiabatic quantum computers race to suppress by ramping slowly enough. Semiclassical WKB quantization, meanwhile, remains the tool of choice for highly excited Rydberg states, molecular vibrational ladders, and tunnelling rates, where Bohr–Sommerfeld with its Maslov $\tfrac12$ nails the levels without the full Schrödinger equation.

The physical picture

Imagine a child on a swing whose chain you shorten by a hair on each pass. Over any single swing the chain barely moves, so the motion is nearly periodic and it makes sense to ask: averaged over one cycle, what survives? The answer is the action — the area the motion sweeps out in the phase plane. The slow squeezing pumps energy in and drives the amplitude up, yet in exactly the proportion that keeps that area fixed. The first figure is the whole story: raise the frequency slowly and the phase-space ellipse turns tall and narrow while enclosing precisely the same area.

Figure

Raising the frequency slowly from ω = 1 to ω = 2 lifts the oscillator's energy from E = 1 to E = 2, yet the phase-space area 2πJ = 2πE/ω stays pinned at 2π: the orbit reshapes from a circle into a tall narrow ellipse of equal area, so the action J = E/ω = 1 is the adiabatic invariant

Secular terms are the symptom of asking the wrong question. Expand $\cos\!\big(\omega(\varepsilon)\,t\big)$ in powers of $\varepsilon$ and you generate $\varepsilon t\sin(\omega_0 t)$ and worse — bounded functions in truth, mangled into unbounded ones by a Taylor series that has no business expanding a phase. Averaging refuses to expand the phase; it tracks the phase exactly and averages only its slow envelope. The same discipline works in space: in the WKB region the local wavelength $2\pi\varepsilon/\sqrt Q$ contracts wherever the "momentum" $\sqrt Q$ is large, and conservation of a local current — amplitude squared times wavenumber — forces the amplitude up as $Q^{-1/4}$ so that ever more tightly packed crests carry the same flux. Push toward a turning point, where $Q\to 0$ , and the wavelength blows up, the amplitude diverges, fast and slow merge, and averaging surrenders to the Airy connection formulae of the neighbouring article.

Figure

In the classically allowed region Q > 0 the local wavelength 2πε/√Q stretches and the amplitude swells as 1/Q^(1/4) toward the turning point, keeping the current amplitude² × √Q constant; where Q → 0 the dashed Q^(−1/4) envelope diverges and WKB fails, handing over to the Airy connection formulae, beyond which the solution decays exponentially into the forbidden region Q < 0

The core equations

Averaging begins with a system split into a fast angle and a slow variable: $\theta$ winds at an $O(1)$ rate while $x$ inches along at rate $\varepsilon$ , and the whole method lives in that separation of speeds.

\dot{x} = \varepsilon\, f(x,\theta),\qquad \dot{\theta} = \omega(x) \qquad (1)

where: $x$ — the slow variable (amplitude, action, guiding centre), drifting at rate $\varepsilon$ ; $\theta$ — the fast phase, $2\pi$ -periodic in $f$ ; $\omega(x)$ — the instantaneous frequency, $O(1)$ ; $\varepsilon$ — the slowness parameter, $0<\varepsilon\ll1$ . Because $f$ is periodic in $\theta$ , its net effect over one turn is its mean. The averaged equation keeps only that mean. It is autonomous — the fast angle has vanished — and Bogoliubov's theorem certifies its solution shadows the true $x$ to $O(\varepsilon)$ for all $t$ up to $O(1/\varepsilon)$ .

\dot{\bar x} = \varepsilon\,\bar f(\bar x),\qquad \bar f(x)=\frac{1}{2\pi}\int_0^{2\pi} f(x,\theta)\,\mathrm d\theta \qquad (2)

where: $\bar x$ — the smoothed slow variable; $\bar f$ — the perturbation averaged over one fast period. The discarded zero-mean fluctuation is precisely the piece naive perturbation theory mishandles into secular growth. The workhorse test case is the weakly nonlinear oscillator. Writing $x=a\cos\psi$ with $\psi=t+\phi$ turns it into a slow equation for the amplitude alone.

\ddot x + x = \varepsilon\, g(x,\dot x),\qquad \dot a = -\frac{\varepsilon}{2\pi}\int_0^{2\pi} g(a\cos\psi,-a\sin\psi)\,\sin\psi\,\mathrm d\psi \qquad (3)

where: $g$ — a weak nonlinear or dissipative force; $a$ — the slow amplitude; $\psi$ — the fast phase, averaged out. The companion equation for $\dot\phi$ gives the frequency shift, which vanishes for van der Pol. Let the oscillator's parameters drift slowly and averaging produces the crown jewel of the subject: the action is an adiabatic invariant, conserved far better than the naive $O(\varepsilon)$ estimate suggests.

J=\frac{1}{2\pi}\oint p\,\mathrm d q = \text{const}+O\!\big(e^{-c/\varepsilon}\big) \qquad (4)

where: $J$ — the action, the phase-space area over $2\pi$ ; $p,q$ — canonical momentum and coordinate; $c$ — a positive constant fixed by the nearest complex singularity of the slow drive. The algebraic corrections all cancel for analytic parameters, leaving only a non-perturbative remainder. The spatial incarnation is the WKB (Liouville–Green) equation — the time-independent Schrödinger equation with $\varepsilon=\hbar$ , and the short-wavelength limit of acoustics and optics.

\varepsilon^2 y'' + Q(x)\,y = 0 \qquad (5)

where: $\varepsilon$ — the semiclassical small parameter; $Q(x)$ — a slowly varying coefficient; $Q>0$ gives an oscillatory (allowed) region and $Q<0$ an exponential (forbidden) one, meeting at turning points $Q=0$ . Its asymptotic solution carries the eikonal phase and the transport amplitude — the spatial echoes of the fast phase and the conserved action.

y(x)\sim \frac{C_\pm}{Q^{1/4}}\,\exp\!\left(\pm\frac{\mathrm i}{\varepsilon}\int^x\!\sqrt{Q(x')}\,\mathrm d x'\right) \qquad (6)

where: $Q^{1/4}$ — the transport amplitude enforcing flux conservation; $\int\!\sqrt Q\,\mathrm dx'$ — the eikonal phase, the classical action over $\varepsilon$ ; $C_\pm$ — constants set by boundary or connection conditions. Amplitude squared times wavenumber is constant, the spatial adiabatic invariant. Closing the two WKB branches across a pair of soft turning points quantizes the bound states, the semiclassical bridge to the old quantum theory.

\oint p\,\mathrm d q = 2\pi\hbar\left(n+\tfrac12\right),\qquad n=0,1,2,\dots \qquad (7)

where: $n$ — the quantum number; the $\tfrac12$ — the Maslov correction, a quarter-wave phase at each of the two soft turning points; $\hbar$ — the reduced Planck constant. The action is now quantized in units of $2\pi\hbar$ . Finally, the tiny non-conservation of the action is set not by any power of $\varepsilon$ but by an exponential of $1/\varepsilon$ — the classical cousin of Landau–Zener leakage.

\delta J \sim \exp\!\left(-\frac{1}{\varepsilon}\,\mathrm{Im}\!\oint_{\mathrm{c.p.}}\!\omega\,\mathrm d t\right) \qquad (8)

where: $\delta J$ — the non-perturbative change of the invariant; $\mathrm{c.p.}$ — the complex turning point of $\omega(t)$ nearest the real axis. The larger that gap, the more perfectly $J$ is frozen.

Derivation AThe averaging theorem from a near-identity transform

The goal is to prove that the cycle average is not a plausible guess but the forced leading-order drift. We construct a periodic, near-identity change of variables to absorb the fast fluctuation of $f$ ; the requirement that the corrector stay bounded and periodic is a solvability condition, met in exactly one way — when the residual drift equals the mean of $f$ . That single condition is the averaging theorem, and it explains why the discarded piece is the very part that would otherwise have grown secularly.

Step 1. Posit $x=\bar x+\varepsilon\,u(\bar x,\theta)$ with $u$ chosen $2\pi$ -periodic in the fast angle $\theta$ , its job being to soak up the oscillatory part of $f$ while $\bar x$ carries only the smooth drift. Why: a change of variables that is the identity to leading order cannot alter the physics, but it gives us a free periodic function $u$ to cancel the troublesome fluctuation.

Step 2. Differentiate and use $(1)$ ; to first order $\dot{\bar x}+\varepsilon\,\omega\,\partial_\theta u=\varepsilon f(\bar x,\theta)+O(\varepsilon^2)$ , and we demand that the smooth part obey $\dot{\bar x}=\varepsilon\bar f(\bar x)$ for some drift $\bar f$ yet to be pinned down. Why: splitting the equation into a smooth drift plus a purely oscillatory remainder is the only way to keep $\bar x$ slow and $u$ bounded.

Step 3. The remainder reads $\omega\,\partial_\theta u=f(\bar x,\theta)-\bar f(\bar x)$ , and a periodic $u$ exists iff the right-hand side has zero average over $\theta$ — the Fredholm solvability condition — which forces $\bar f(\bar x)=\tfrac{1}{2\pi}\int_0^{2\pi}f\,\mathrm d\theta$ . Why: integrating a periodic $\partial_\theta u$ over a full period gives zero, so any non-zero mean would demand a non-periodic (secularly growing) $u$ — exactly the pathology we set out to avoid.

Step 4. With $\bar f$ fixed as the mean, $u$ is the bounded antiderivative of the fluctuation, and Bogoliubov's theorem upgrades the formal cancellation to a uniform $O(\varepsilon)$ bound on $|x-\bar x|$ for $t$ up to $O(1/\varepsilon)$ . Why: the near-identity map has bounded inverse, so smallness of the averaged error transfers directly to the true solution over the entire slow time scale.

Go deeper · Derivation B · 1 minAdiabatic invariance of the slowly varying harmonic oscillator

Derivation BAdiabatic invariance of the slowly varying harmonic oscillator

The aim is to show that for $\ddot x+\omega^2(\varepsilon t)\,x=0$ with a slowly changing frequency the ratio $E/\omega$ is conserved and equals the action. We solve the oscillator by WKB in time, compute the cycle-averaged energy, form the ratio, watch the slow terms cancel, and read off the connection to $\oint p\,\mathrm dq$ and to the quantum number $n$ — closing the loop between classical adiabatic invariance and the quantum adiabatic theorem.

Step 1. Take the temporal Liouville–Green form $x=C\,\omega^{-1/2}\cos\Theta$ with $\Theta=\int^t\omega\,\mathrm dt'$ ; the $\omega^{-1/2}$ prefactor is the transport amplitude, the temporal twin of $Q^{-1/4}$ . Why: this is the WKB solution of the oscillator, valid because $\omega$ changes little over one period, packaging the fast oscillation into $\Theta$ and the slow drift into the amplitude.

Step 2. Differentiate: $\dot x=-C\,\omega^{1/2}\sin\Theta+O(\varepsilon)$ , since $\tfrac{\mathrm d}{\mathrm dt}\omega^{-1/2}$ is itself $O(\varepsilon)$ ; then $E=\tfrac12\dot x^2+\tfrac12\omega^2x^2=\tfrac12C^2\omega\,(\sin^2\Theta+\cos^2\Theta)=\tfrac12C^2\omega$ . Why: the cross term $\propto\dot\omega/\omega$ is first order in slowness and averages to zero, so the fast oscillation drops out and only the smooth $\tfrac12C^2\omega$ survives.

Step 3. Form the ratio $E/\omega=\tfrac12C^2$ , a constant because $C$ is; and since for a harmonic oscillator $\oint p\,\mathrm dq=2\pi E/\omega$ , the invariant is exactly the action $J=E/\omega$ . Why: dividing out the one slowly varying factor $\omega$ isolates the genuinely conserved combination, and identifying it with the phase-space area shows it is the same $J$ that averaging conserves in general.

Step 4. The quantum echo is immediate: $E_n=\hbar\omega(n+\tfrac12)$ gives $E_n/\omega=\hbar(n+\tfrac12)$ , constant precisely when the quantum number $n$ does not change. Why: the classical adiabatic invariant equals $\hbar$ times a fixed integer, which is the statement that a slow change keeps the system in its instantaneous energy level — the quantum adiabatic theorem.

Go deeper · Derivation C · 1 minWKB eikonal and transport, and the meaning of the quarter-power

Derivation CWKB eikonal and transport, and the meaning of the quarter-power

The purpose here is to extract both the phase and the amplitude of the WKB solution from a single exponential ansatz, then give the $Q^{-1/4}$ prefactor its physical name by an independent flux argument. We substitute a phase expansion into $(5)$ , solve order by order in $\varepsilon$ — eikonal at leading order, transport at the next — and a Wronskian current then reproduces the same amplitude as a conservation law, showing the two routes are one.

Step 1. Write $y=\exp\!\big(\tfrac1\varepsilon\sum_{n\ge0}\varepsilon^n S_n(x)\big)$ and insert into $(5)$ ; the leading $O(\varepsilon^0)$ balance is $(S_0')^2+Q=0$ , so $S_0=\pm\,\mathrm i\!\int\sqrt Q\,\mathrm dx$ . Why: this is the Hamilton–Jacobi equation for the phase — the eikonal — identifying $S_0'$ with the local momentum $\pm\mathrm i\sqrt Q$ and fixing the fast oscillation.

Step 2. The next order $O(\varepsilon^1)$ gives the transport equation $2S_0'S_1'+S_0''=0$ , so $S_1'=-\tfrac12 S_0''/S_0'=-\tfrac14\,Q'/Q$ and $S_1=-\tfrac14\ln Q$ , i.e. the amplitude $e^{S_1}=Q^{-1/4}$ . Why: once the phase is fixed, the amplitude is not free — it solves a linear first-order equation whose unique solution is the quarter-power that suppresses secular growth.

Step 3. Independently, define the current $j=\tfrac{\varepsilon}{2\mathrm i}(y^*y'-y\,y^{*\prime})$ ; because $y''=-Qy/\varepsilon^2$ with real $Q$ , one finds $\mathrm dj/\mathrm dx=0$ , and the WKB form gives $j\approx\sqrt Q\,|C|^2/\sqrt Q=|C|^2$ . Why: constancy of $j$ demands $|C|^2$ be constant, so the amplitude scales as $Q^{-1/4}$ — the transport equation re-derived as a conservation law.

Step 4. Read it physically: $|y|^2$ times the local wavenumber $\sqrt Q/\varepsilon$ is constant, so probability density times momentum — a local action density — is conserved as the wave threads through the medium. Why: this is $E/\omega$ in space; more tightly packed crests (large $\sqrt Q$ ) must carry proportionally smaller squared amplitude to hold the flux fixed, exactly what $Q^{-1/4}$ arranges.

Limiting cases

How well averaging works is governed by one thing: how cleanly the fast phase outruns the slow variables. Where that separation is sharp the method is superb; where it collapses it fails, as the table's lower rows show.

Case	Result	Meaning
Strongly adiabatic	$\varepsilon\to0,\ t\sim1/\varepsilon$	Averaging is uniformly valid; the action is conserved to $O(\varepsilon)$ over the slow time, and exponentially well for analytic, smoothly switched forcing.
Near resonance	$\omega_1/\omega_2\to p/q$	Small denominators $1/(p\omega_1-q\omega_2)$ wreck the average; one must retain the slow resonant angle $p\phi_1-q\phi_2$ and average over the rest.
Turning point / separatrix	$Q\to0$ or $\omega\to0$	Fast and slow merge; the action jumps as an orbit crosses a separatrix and WKB diverges as $Q^{-1/4}\to\infty$ . Connection formulae replace averaging.
Geometrical-optics limit	$\varepsilon=\hbar\to0$	The phase $S/\hbar$ obeys Hamilton–Jacobi; rays are classical trajectories and the amplitude transports a conserved flux.

Common trapAveraging is always valid because ε is small

Not so — small $\varepsilon$ is necessary but nowhere near sufficient. Averaging rests on the fast phase genuinely racing past the slow variables, and there are two routine ways for that to fail no matter how small $\varepsilon$ is. Near a resonance, where two frequencies become commensurate ( $p\omega_1\approx q\omega_2$ ), the combination $p\phi_1-q\phi_2$ crawls rather than races, small denominators $1/(p\omega_1-q\omega_2)$ blow up the corrections, and one must keep that slow resonant angle explicitly. Near a turning point or separatrix, where the frequency itself vanishes ( $\omega\to0$ , or $Q\to0$ in WKB), the fast period diverges, the scales coincide, $Q^{-1/4}$ blows up, and the invariant can jump by $O(1)$ as the orbit crosses the separatrix. In both cases the fix is not a smaller $\varepsilon$ but a different method: resonant (partial) averaging in the first, connection formulae or separatrix-crossing theory in the second.

Worked examples

1 · The van der Pol limit cycle (fully worked)

Apply amplitude–phase averaging to $\ddot x-\varepsilon(1-x^2)\dot x+x=0$ to find the amplitude equation, the limit-cycle radius, and its stability. Here the perturbation is $g(x,\dot x)=(1-x^2)\dot x$ .

Step 1. Write $x=a\cos\psi$ , $\dot x=-a\sin\psi$ with $\psi=t+\phi$ , so $g=(1-a^2\cos^2\psi)(-a\sin\psi)$ . Equation $(3)$ gives the slow amplitude drift $\dot a=-\dfrac{\varepsilon}{2\pi}\displaystyle\int_0^{2\pi}g\,\sin\psi\,\mathrm d\psi=\dfrac{\varepsilon a}{2\pi}\displaystyle\int_0^{2\pi}(1-a^2\cos^2\psi)\sin^2\psi\,\mathrm d\psi$ .

Step 2. Evaluate the two cycle averages that appear: $\langle\sin^2\psi\rangle=\tfrac12$ and $\langle\cos^2\psi\sin^2\psi\rangle=\tfrac18$ . Substituting, $\dot a=\varepsilon a\big(\tfrac12-\tfrac18 a^2\big)=\dfrac{\varepsilon}{2}\,a\Big(1-\dfrac{a^2}{4}\Big)$ . The companion phase equation averages to $\dot\phi=0$ , so there is no frequency shift at $O(\varepsilon)$ .

Step 3. Set $\dot a=0$ . The trivial root $a=0$ is the (unstable) rest state; the non-trivial root solves $1-a^2/4=0$ , giving the limit-cycle radius $a_\infty=2$ , independent of $\varepsilon$ .

Step 4. Test stability by perturbing $a=2+\delta a$ : linearising gives $\dot{\delta a}=-\varepsilon\,\delta a$ , an exponential decay for $\varepsilon>0$ , so the cycle is stable — any start (say $a=0.1$ ) spirals monotonically out to $a=2$ . This reproduces the standard van der Pol result (Bender & Orszag §11.2) without ever integrating the full ODE. Amplitude equation $\dot a=\tfrac{\varepsilon}{2}a(1-a^2/4)$ , stable limit cycle at $a=2$ .

2 · Slow compression of an oscillator (the Solvay pendulum, made numerical)

A unit-mass oscillator ( $m=1\ \mathrm{kg}$ ) has frequency $\omega_1=1\ \mathrm{rad/s}$ and amplitude $A_1=\sqrt2\ \mathrm{m}$ . The frequency is then raised very slowly to $\omega_2=2\ \mathrm{rad/s}$ — as if a pendulum's string were reeled in to a quarter of its length. Find the new energy, amplitude, action, and phase-space area.

Step 1. Initial energy: $E_1=\tfrac12 m\,\omega_1^2 A_1^2=\tfrac12(1)(1)^2(\sqrt2)^2=\mathbf{1\ J}$ . The action is $J=E/\omega$ , so $J_1=1/1=\mathbf{1\ J\cdot s}$ , and the phase-space area is $2\pi J_1=\mathbf{2\pi}$ .

Step 2. Because the change is slow, the action is the adiabatic invariant: $J_2=J_1=1\ \mathrm{J\cdot s}$ . Hence the energy tracks the frequency, $E_2=J\,\omega_2=1\times2=\mathbf{2\ J}$ — the slow squeezing has pumped in exactly one joule, doubling the energy just as it doubled the frequency.

Step 3. The new amplitude follows from $E_2=\tfrac12 m\,\omega_2^2 A_2^2$ : $A_2=\sqrt{2E_2/(m\omega_2^2)}=\sqrt{4/4}=\mathbf{1\ m}$ . The swing has shrunk from $\sqrt2\approx1.41$ m to $1$ m even though the energy rose — the stiffer, faster oscillator stores more energy in a smaller excursion.

Step 4. Check the invariant directly: phase-space area $2\pi J_2=2\pi E_2/\omega_2=2\pi(2)/2=\mathbf{2\pi}$ , identical to Step 1 — the enclosed area never moved, precisely the first figure. E: 1 → 2 J, amplitude: √2 → 1 m, action J = E/ω = 1 J·s and area 2π both conserved.

Physics of the portrait

The adiabatic invariant is the thread that ties this whole chapter together — a quantity that survives when everything around it changes slowly, discovered in classical mechanics, elevated into the founding principle of the old quantum theory, and finally proven to be conserved with almost miraculous, exponential precision.

Origins · 1 minadiabatic invariant and how this was discovered

adiabatic invariant

the action J = ∮p dq, conserved under slow change of parameters · classical averaging, WKB and semiclassical mechanics

The idea that some combination of an oscillator's variables should be immune to slow change is older than quantum mechanics, but it crystallised at the very first Solvay Congress in 1911, when Hendrik Lorentz posed a puzzle to the founders of quantum theory: if you slowly shorten the string of a swinging quantum pendulum, what happens to its quantum of energy? Albert Einstein answered at once — energy and frequency change in lockstep, so their ratio $E/\nu$ stays fixed. Paul Ehrenfest seized on that ratio. Between 1913 and 1917 he built his "adiabatic hypothesis" around it: the only quantities that may sensibly be quantized are those that are adiabatically invariant, because only they survive the slow deformations connecting one situation to another. This gave the old quantum theory its rule for which classical actions $\oint p\,\mathrm dq$ to set equal to integer multiples of $h$ . The classical machinery underneath, sharpened by Johannes Burgers, was made rigorous and startlingly strong decades later by Vladimir Arnold and Martin Kruskal, who showed the invariant leaks only exponentially, $\delta J\sim e^{-c/\varepsilon}$ , for analytic drives. What Einstein sketched in an afternoon became a theorem: slowness plus smoothness freezes the action almost perfectly.

History

1837

— Liouville and Green find the Q^(−1/4) amplitude for a slowly varying oscillator

1911

— at the Solvay Congress the quantum-pendulum question fixes E/ν as the invariant

1917

— Ehrenfest formalises the adiabatic hypothesis of the old quantum theory

1926

— Wentzel, Kramers and Brillouin build the semiclassical method, after Jeffreys in 1923

1937

— Krylov and Bogoliubov formalise averaging; Arnold and Kruskal later prove its exponential accuracy

Going further

Texts. Bender & Orszag, Advanced Mathematical Methods for Scientists and Engineers (1978), ch. 10 (WKB) and ch. 11 (averaging, multiple scales) · Stone & Goldbart, Mathematics for Physics (CUP 2009), asymptotics and WKB chapters · Arfken, Weber & Harris, Mathematical Methods for Physicists, 7th ed. · Hassani, Mathematical Physics, 2nd ed. · Landau & Lifshitz, Mechanics, §49 (adiabatic invariants) · Arnold, Mathematical Methods of Classical Mechanics, ch. 10 (averaging and adiabatic invariants) · Byron & Fuller, Mathematics of Classical and Quantum Physics, on asymptotic methods.

Provenance. The averaging theorem and near-identity transform follow Bogoliubov–Mitropolsky and Arnold ch. 10; the invariant $E/\omega$ and $J=\oint p\,\mathrm dq$ are Landau–Lifshitz §49; the WKB transport solution $S_1=-\tfrac14\ln Q$ and the flux argument are Bender & Orszag ch. 10 and Stone & Goldbart. A verifier should reconfirm: the van der Pol averages $\langle\sin^2\rangle=\tfrac12$ , $\langle\cos^2\sin^2\rangle=\tfrac18$ giving limit-cycle radius $a=2$ and stability $\dot{\delta a}=-\varepsilon\,\delta a$ ; the compression numbers $E_1=1\to E_2=2$ J, $A_1=\sqrt2\to A_2=1$ m, $J=1$ J·s and area $2\pi$ both invariant; the amplitude $Q^{-1/4}$ ; and the Maslov $\tfrac12$ . All are standard textbook results.

Prerequisites. Asymptotic Series and Poincaré Expansions · Order Symbols and Gauge Functions · The WKB Approximation. Leads to. Connection Formulae and Turning Points · Bohr–Sommerfeld Quantization · The Method of Stationary Phase.

Ask the tutor: Ask me to average any weakly perturbed oscillator, or to walk a slowly varying one through its adiabatic invariant and its WKB twin step by step.

What actually goes wrong with ordinary perturbation theory here?

It generates secular terms — contributions growing like $\varepsilon t$ , $\varepsilon^2 t^2$ , and so on. The culprit is that the true frequency depends on $\varepsilon$ , and expanding an oscillation of shifted frequency in a Taylor series produces $t\sin t$ pieces that are bounded in reality but unbounded term by term. Over short times the series is fine, but at $t\sim1/\varepsilon$ — the very scale where the slow physics happens — the "correction" is as large as the leading term and the ordering collapses. Averaging never expands the phase: it keeps the phase exact and resums the dangerous series into a slowly drifting amplitude or action.

Is the action conserved exactly, or only approximately?

Neither naive answer is right. The action is not exactly conserved, but far better than "to order $\varepsilon$ " — every algebraic power of $\varepsilon$ cancels for a smooth, analytic drive, and the leftover change is exponentially small, $\delta J\sim e^{-c/\varepsilon}$ , with $c$ set by the nearest complex singularity of the slow parameter. So over the slow time scale it is conserved to astonishing precision. The one important exception is a separatrix: sweep the orbit across one (say a pendulum driven from libration to rotation) and the action can jump by $O(1)$ , because there the period diverges and the adiabatic assumption fails outright.

Why is the WKB approximation the same idea as averaging?

Because both are two-scale methods with "fast" and "slow" merely relabelled. In averaging the fast variable is time-like — a phase $\theta$ that winds quickly — while amplitude and action drift slowly. In WKB it is space-like: the eikonal phase $\tfrac1\varepsilon\int\sqrt Q$ oscillates rapidly while the local wavenumber $\sqrt Q$ and the amplitude vary gently. Demanding no secular growth fixes the amplitude to $Q^{-1/4}$ , the statement that a local action density — amplitude squared times wavenumber — is conserved, exactly as the temporal action $E/\omega$ is. Same theorem, transposed from time to space.

Where does the one-half in Bohr–Sommerfeld come from?

From the turning points. A pure phase-integral quantization would read $\oint p\,\mathrm dq=2\pi\hbar n$ , but that ignores what happens where the WKB wave reflects. At each soft (linear) turning point the connection formulae show the wave picks up an extra quarter-wavelength of phase — a $\pi/2$ shift, the Maslov index. A bound state between two such turning points collects two of them, $2\times\pi/2=\pi$ , which is half a period, and that half shifts the quantization condition to $\oint p\,\mathrm dq=2\pi\hbar(n+\tfrac12)$ . It is precisely why the WKB harmonic-oscillator levels come out exactly right, zero-point energy included, even though WKB is only an approximation.

What is an adiabatic invariant, physically?

It is an area. For a one-dimensional oscillator the action $J=\tfrac1{2\pi}\oint p\,\mathrm dq$ is the phase-plane area enclosed by the orbit, divided by $2\pi$ . Change a parameter slowly and the orbit deforms — stretching, tilting, swelling in energy — but always in the way that preserves the area it sweeps: a slow, smooth deformation is an area-preserving map on the family of orbits. Because phase-space area is what quantum mechanics measures in units of $2\pi\hbar$ , the classical adiabatic invariant is also what keeps a quantum system in a fixed energy level under slow driving.

What happens to WKB at a turning point?

At a turning point $Q(x)=0$ the wavenumber $\sqrt Q$ vanishes, the wavelength diverges, and the amplitude $Q^{-1/4}$ blows up — nonsense for a physical wave, and the signal that fast and slow scales have merged. The cure is to zoom in: near a linear turning point $(5)$ reduces to Airy's equation, whose exact solution smoothly bridges the oscillatory and exponential sides. Matching the WKB forms onto it is the connection-formula machinery of the next article, and where the Maslov $\tfrac12$ is born.