WKB approximation

For a large-parameter second-order ODE, WKB separates a rapidly varying exponential phase from a slowly varying transport amplitude.

For a real scalar wave equation $E^{\prime \prime }+k^{2}E=0$, the next order of the WKB approximation gives $2k{a}^{\prime }+k^{\prime }a=0$. Thus $a=A/\sqrt{k}$, and the leading Wronskian flux of a right-travelling component is $|A{|}^2$ independently of the local wavenumber.

A zero of the local squared wavenumber in $E^{\prime \prime }+k^2(x)E=0$. At a simple zero, $(k^2{)}^{\prime }(x_T)\ne 0$, the WKB approximation fails in a layer with scale $|(k^2{)}^{\prime }(x_T){|}^{-1/3}$ and the Airy ordinary differential equation supplies the local replacement. Two nearby wave turning points require a uniform treatment rather than independent Airy turning-point connection formulas.

At fixed incident energy and fixed potential-energy profile, the forbidden-region WKB action is $\int \sqrt{2m(V-E)}dx$. Its leading transmission exponent grows as the square root of mass. Equal electrostatic voltages need not define equal potential-energy barriers for particles with different charges. The leading exponential comparison assumes a sufficiently opaque barrier; prefactors and the WKB turning-point conditions remain relevant.

For an opaque barrier between simple turning points, the leading transmission probability is $T\simeq e^{-2S_B/\hslash }$, where $S_B=\int \sqrt{2m[V(x)-E]}\mathrm{d}x$ over the forbidden interval. The exponent for probability is twice that for amplitude. Matching determines prefactors and the domain where this approximation applies.

For $p(x)=\sqrt{2m[E-V(x)]}$ in a classically allowed region, a local validity condition is $|\hslash p^{\prime }/p^2|\ll 1$. The momentum changes little during a local oscillation. It fails at an ordinary turning point where $p=0$, requiring an Airy turning-point connection formula.

For a slowly varying oscillator with rapidly decaying forcing, first construct the localized particular solution $u_p$. Apply the initial conditions to the residual $u_h$ to determine its WKB approximation amplitude and phase constants. Ignoring the early forcing because it later becomes small loses the permanent free oscillations it excites. In $u^{\prime \prime }+4e^{-2\epsilon t}u=8e^{-2t}$, the particular expansion is $u_p=e^{-2t}[1+\epsilon (t+1/2)]+O({\epsilon }^2)$, whose initial value shifts the first homogeneous cosine coefficient by $-\epsilon /2$.

For $A_t=-a(ϵx)A_x-ia(ϵx{)}^2{A}_{xx}$, a downstream WKB approximation at fixed frequency $\Omega <1/4$ has local wavenumber $k=q/a$. The next-order amplitude equation is $(a-2a^{2}k)B_X=a^2{k}_{X}B$, giving the stated envelope. For small $\Omega$, it reduces to $B\propto a^{-\Omega }$ to first order in frequency. The approximation requires positive slowly varying width and separation from the double root at $\Omega =1/4$.

The Liouville transformation of a second-order equation with $t={\int }^x\sqrt{f}dx$ gives $\ddot{U}-(1+F)U=0$. Neglecting this correction yields amplitude $f^{-1/4}$ and phase or exponent $\int \sqrt{f}$. Its smallness measures a local slowly-varying-coefficient approximation, away from zeros of $f$.

For ${ϵ}^2{y}_{XX}+k^2(X)y=0$ with nonzero slowly varying $k$,It fails near turning points and where $ϵ|k^{\prime }|/|k{|}^2$ is not small.

For smooth $K=|k|>0$ and fixed amplitude flux, use $\sigma =K+{ϵ}^2{\sigma }_1+\cdots$ and $A_0\propto K^{-1/2}$. The exact amplitude-phase equation gives ${\sigma }_1=A_0^{\prime \prime }/(2K{A}_0)$, which is the displayed formula. The first phase correction is order ${ϵ}^2$ in the local wavenumber, hence order $ϵ$ in its integrated exponential phase.

An oscillator with frequency $\omega (t)=e^{-\epsilon t}$ leaves the WKB approximation when $|{\omega }_t|/{\omega }^2=\epsilon e^{\epsilon t}$ becomes order one. The shifted variable $T=\epsilon t-\log ({\epsilon }^{-1})$ changes its leading undamped equation to $Y_{TT}+e^{-2T}Y=0$. With $Z=e^{-T}$ this is the order-zero Bessel differential equation. Large-$Z$ Bessel functions match the earlier oscillations, while the small-$Z$ constant and logarithmic solutions become a constant and a linear function of late time.

For $y_{tt}+e^{-2\epsilon t}(\epsilon y_t+y)=0$, $y(0)=0$, $y_t(0)=1$, set $r=e^{-2\epsilon t}/2$. The exact equation becomes $r{y}_{rr}+(1-r)y_r+y/(2{\epsilon }^2)=0$. Let $m(t)=M(-1/(2{\epsilon }^2),1,r)$, using the Kummer function. It is regular at $r=0$ and tends to one. The Wronskian $W=m{y}_t-m_{t}y$ obeys $W_t=-\epsilon e^{-2\epsilon t}W$, with $W(0)=M(-1/(2{\epsilon }^2),1,1/2)$. Integration gives the displayed exact late slope. The other local solution grows at most logarithmically in $r$, so $m_{t}y\to 0$ and $W\to y_t$. This provides a check on a WKB approximation even near phases where its leading predicted slope vanishes.

For a separated duct mode with local width $h(X)$ and nonzero axial wavenumber $k(X)$, conservation of modal flux gives amplitude proportional to $[h(X)k(X){]}^{-1/2}$. The phase is ${ϵ}^{-1}{\int }^{X}k(\xi )d\xi$.

For $\nabla \cdot (k_0(X{)}^{-2}\nabla p)+p=0$ in a slowly varying rigid duct, a real local transverse normal mode $F$ with squared norm $N(X)$ has longitudinal wavenumber $k(X)$. The WKB approximation is $p=A(X)F(y,X)\exp [-i{ϵ}^{-1}{\int }^{X}k(\xi )d\xi ]$. The moving-wall Neumann boundary condition and next-order solvability give $2kN{k}_0^{-2}{A}_X+(kN{k}_0^{-2}{)}_{X}A=0$, hence the displayed amplitude. For $F=\cos (n\pi y/R)$, $N=R$ for $n\ge 1$ and $N=2R$ for $n=0$. The factor $k_0$ is essential when the coefficient in the divergence-form equation varies. This approximation requires a mode separated from cutoff and from other modes.

Near a simple turning point, rescaling the independent variable reduces a second-order equation to the Airy equation. The decaying Airy solution on the forbidden side matches an oscillatory solution with phase shift $\pi /4$ on the allowed side.

For a one-dimensional energy $E$ and potential $V(x)$, a classical turning point satisfies $V(x)=E$. The classically allowed region has $E>V(x)$ and supports an oscillatory leading-order WKB solution.

For an oscillatory interval terminated by a hard boundary and a simple turning point, matching the boundary condition to the decaying solution gives an action integral equal to an integer multiple of $\pi$ with the turning-point phase correction.

For a smooth symmetric confining potential energy with four separated simple classical turning points $-x_2<-x_1<x_1<x_2$, defineThe Airy turning-point connection formula applied to a decaying exterior wavefunction gives a central WKB approximation proportional to ${\chi }^{-1/2}[2\cos \theta e^K+\sin \theta e^{-K}]$, where $\chi =\sqrt{2m(V-E)}$ and $K(x)={\hslash }^{-1}{\int }_x^{x_1}\chi dx$. Since ${\chi }^{\prime }(0)=0$ and $2K(0)=\varphi$, the even Neumann boundary condition gives $\tan \theta =2e^{\varphi }$, while the odd Dirichlet boundary condition gives $\tan \theta =-2e^{\varphi }$. For $\varphi \gg 1$, $\theta =(n+\frac{1}{2})\pi \mp \frac{1}{2}{e}^{-\varphi }+O(e^{-3\varphi })$, with the minus correction for even parity. Since ${\theta }^{\prime }=\mathcal{T}/(2\hslash )$ for the single-well classical oscillation period $\mathcal{T}$, the leading double-well tunneling splitting is $2\hslash e^{-\varphi }/\mathcal{T}$.

A Dirichlet boundary condition at a hard endpoint and a single smooth classical turning point give the leading WKB quantization condition ${\int }_0^{x_{\mathrm{t}}}k(x)dx=(n+3/4)\pi$, with $n=0,1,\dots$. The phase differs from the two-soft-turning-point condition because the hard wall imposes an exact wavefunction zero rather than an Airy function connection phase. The WKB validity condition is still checked in the interior.

For the half-line square-root potential well, the dimensionless action integral isHence $E_n/E_{\ast }=[15\pi (n+3/4)/8{]}^{2/5}$. The ground-state estimate is $1.811694E_{\ast }$. This is a semiclassical estimate, not a variational upper bound.

For smooth positive $K$ on an interval with regular Dirichlet boundary conditions, there is no turning-point phase shift. Combining conjugate WKB approximation branches to vanish at the first endpoint makes the total phase at the second an integer multiple of $\pi$. The first correction is the integrated WKB phase correction for a slowly varying oscillator. For $K=x+1$ on $[0,1]$ the resulting eigenvalues are $4{\pi }^2{n}^2/9-3/16+O(n^{-2})$.

For two simple turning points $a(E)<b(E)$ surrounding one classically allowed well,

The semiclassical action integralis strictly increasing for a confining one-dimensional potential. Consequently the two-turning-point WKB quantization condition has at most one energy at each quantum number.

For the even potentialthe turning points are $\pm 4E$ for $0\le E\le 1$ and $\pm (E+1{)}^2$ for $E\ge 1$. Its action is

Forthe substitutiongives the normal form

To classify infinity, set $z=1/x$ and classify $z=0$. For a second-order equation in normal form, a transformed coefficient with a pole of order greater than two makes infinity an irregular singular point.

Writing $v=e^S$ in $v^{\prime \prime }=q(x)v$ produces the Riccati equationExpanding $S^{\prime }$ in inverse powers of $x$ determines the exponential phase and algebraic prefactor recursively.

A Riccati equation is a first-order equation quadratic in the unknown function.

Compare the displayed ordinary differential equation with $u^{\prime }=1+u^2$, $u(0)=1$, whose solution is $u=\tan (x+\pi /4)$. Their difference satisfies $w^{\prime }=(y+u)w+x^2$, $w(0)=0$. The integrating factor formula makes $w\ge 0$ on the common forward existence interval. Since $u$ diverges at $\pi /4$, the increasing solution $y$ has a maximal forward endpoint $0<x_{\ast }\le \pi /4$. If $y$ remained bounded at this finite endpoint, local existence with the smooth right side would continue it, a contradiction. Hence $y\to +\infty$. The Picard-Lindelöf theorem guarantees local existence, not existence through such a finite-time blow-up. On $|x|\le 1$, $|y|\le 10$, a bound $|1+x^2+y^2|\le 102$ and distance nine to the nearer horizontal box edge give the explicit guaranteed interval $|x|\le 3/34$.

Three distinct particular solutions $g,h,j$ of a Riccati equation determine every solution by one constant cross-ratio. Indeed, for $f^{\prime }=a{f}^2+bf+c$, differences satisfy $(f-g{)}^{\prime }=(a(f+g)+b)(f-g)$. The logarithmic derivatives cancel in the displayed cross-ratio, so it is constant. Solving for $f$ gives a rational function of the three solutions and $K$, interpreted on the Riemann sphere at poles.

For constant nonzero $c$, putting $y=x^{\prime }/(cx)$ in $y^{\prime }+c{y}^2+a(t)y+b(t)=0$ cancels the quadratic derivative terms and gives $x^{\prime \prime }+a(t)x^{\prime }+cb(t)x=0$. The logarithmic derivative is defined on intervals where $x\ne 0$, and multiplying $x$ by a nonzero scalar does not change $y$. For variable $c(t)\ne 0$, the derivative coefficient becomes $a-c^{\prime }/c$.

The linearization of a Riccati equation turns this equation into $z^{\prime \prime }-(a+b)z^{\prime }+abz=0$. For distinct real roots, every nonzero linearizing solution is $z=A{e}^{ax}+B{e}^{bx}$ and $y=(aA{e}^{ax}+bB{e}^{bx})/(A{e}^{ax}+B{e}^{bx})$ on intervals with nonzero denominator. The constants $y=a,b$ arise when one coefficient vanishes. With $A\ne 0$, the relative coefficient $B/A$ can have either sign; replacing it by a fixed-sign real exponential would omit a real family of solutions.

For $a=b$, the solutions are $z=(A+Bx)e^{ax}$ and $y=a+B/(A+Bx)$. The equilibrium $y=a$ occurs at $B=0$. Zeros of a nontrivial $z$ give poles of a Riccati solution from zeros of its linearizing solution, so the corresponding $y$ is defined separately between them. Multiplying both coefficients by the same nonzero scalar changes $z$ but not $y$.

At a regular point of a second-order linear ordinary differential equation, a nontrivial solution cannot have both $x$ and $x^{\prime }$ zero, by initial-value uniqueness. Its zeros are therefore simple. Under the linearization of a Riccati equation, such a zero $t_{\ast }$ produces $y=x^{\prime }/(cx)\sim 1/[c(t-t_{\ast })]$. Zeros separate the intervals on which the Riccati equation solution is finite.