WKB in Classically Forbidden Regions

In a classically forbidden region, the potential exceeds the energy and the local classical momentum is imaginary. The oscillatory WKB branches analytically continue into real growing and decaying exponentials with a slowly varying factor $\kappa^{-1/2}$.

Required background. WKB approximation derives the local phase and transport equations; boundary conditions select their physical combination. Turning-point matching connects these branches to an allowed region.

The Local Evanescent Basis

For the stationary Schrödinger equation

$$\left[ - \frac{\hbar^2}{2m} \frac{d^2}{dx^2} + V(x) \right]\psi(x) = E\psi(x),$$

consider an open interval $I$ on which

$$V(x)-E\gt0.$$

Define the positive under-barrier momentum magnitude

$$\kappa(x) = \sqrt{2m\bigl(V(x)-E\bigr)}.$$

Here $\kappa$ has units of momentum. The inverse decay length is

$$q(x) = \frac{\kappa(x)}{\hbar}.$$

Keeping these conventions separate prevents a common missing-$\hbar$ error.

Choose a reference point $x_*\in I$ and define

$$\eta(x;x_*) = \frac{1}{\hbar} \int_{x_*}^{x} \kappa(s)\,ds.$$

Because $\kappa\gt0$, $\eta$ increases with $x$. The leading WKB basis is

$$\begin{aligned} w_{\mathrm g}(x) &= \frac{e^{+\eta(x)}}{\sqrt{\kappa(x)}}, \\ w_{\mathrm d}(x) &= \frac{e^{-\eta(x)}}{\sqrt{\kappa(x)}}. \end{aligned}$$

Relative to increasing $x$, $w_{\mathrm g}$ grows and $w_{\mathrm d}$ decays. The general local solution is

$$\psi_{\mathrm{WKB}}(x) \approx D_{\mathrm g}w_{\mathrm g}(x) + D_{\mathrm d}w_{\mathrm d}(x).$$

The labels are orientation-dependent. If the action integral is written from $x$ to a right-hand endpoint, the same exponential may acquire the opposite sign in its displayed exponent. A reliable calculation states the integration limits before calling a branch growing or decaying.

Changing the phase reference

If the reference point changes so that

$$\widetilde\eta(x) = \eta(x)-\beta,$$

then the same wavefunction requires

$$\widetilde D_{\mathrm g} = D_{\mathrm g}e^{\beta}, \qquad \widetilde D_{\mathrm d} = D_{\mathrm d}e^{-\beta}.$$

Individual coefficient sizes therefore depend strongly on the reference convention. Physical boundary amplitudes, currents, and matched observables do not.

With $\eta(x)=\hbar^{-1}\int_{x_*}^{x}\kappa(s)\,ds$, the two local branches scale as $e^{+\eta}$ and $e^{-\eta}$. “Growing” and “decaying” refer to the chosen direction of increasing $\eta$, not to an intrinsic label independent of integration limits.

Boundary Conditions Select the Physical Combination

The local differential equation has two independent branches. A physical problem selects their coefficients globally.

Semi-infinite forbidden tail

Suppose the forbidden region extends to $x\to+\infty$ and

$$\eta(x)\to+\infty.$$

Normalizability requires

$$D_{\mathrm g}=0,$$

leaving

$$\psi(x) \propto \frac{e^{-\eta(x)}}{\sqrt{\kappa(x)}}.$$

This is the familiar decaying tail of a bound state. On a left-hand tail, the integral orientation must be reversed so that the selected branch decays as $x\to-\infty$.

Finite forbidden interval

Inside a finite barrier, neither local branch should be discarded before matching both ends. A solution that is small toward one edge can contain a component that grows toward the other. The coefficients are determined by the incoming, reflected, and transmitted boundary conditions after connection through both turning-point neighborhoods.

This is why the slogan “throw away the growing exponential” has limited scope. It is correct for a semi-infinite normalizable tail. It is generally wrong as a local rule inside a finite barrier.

Bound-state barrier between two wells

In a double well, the central forbidden interval connects two allowed wells. Exponentially small overlap through that interval produces even–odd level splitting. The local basis here supplies the tails; Double-Well Tunneling explains the resulting two-state dynamics and splitting estimate.

Probability Current Under a Barrier

For a real scalar potential,

$$j = \frac{\hbar}{m} \operatorname{Im} \left( \psi^*\frac{d\psi}{dx} \right).$$

Each isolated basis function $w_{\mathrm g}$ or $w_{\mathrm d}$ is real, up to a constant overall phase. Therefore

$$j[w_{\mathrm g}] = j[w_{\mathrm d}] = 0.$$

This does not imply that every stationary state has zero current inside a forbidden region.

For

$$\psi = D_{\mathrm g}w_{\mathrm g} + D_{\mathrm d}w_{\mathrm d},$$

the Wronskian is

$$\begin{aligned} W[w_{\mathrm g},w_{\mathrm d}] &= w_{\mathrm g}w_{\mathrm d}' - w_{\mathrm g}'w_{\mathrm d} \\ &= - \frac{2}{\hbar}. \end{aligned}$$

The current is consequently

$$j_{\mathrm{WKB}} = - \frac{2}{m} \operatorname{Im} \left( D_{\mathrm g}^*D_{\mathrm d} \right).$$

A nonzero current requires both branches and a nontrivial relative complex phase. The sign shown follows the convention that $\eta$ increases with $x$ and that $w_{\mathrm g}=e^{+\eta}/\sqrt\kappa$ is listed first.

The corresponding density is

$$\lvert\psi\rvert^2 ={} \frac{1}{\kappa} \bigg[ \lvert D_{\mathrm g}\rvert^2e^{2\eta} + \lvert D_{\mathrm d}\rvert^2e^{-2\eta} + 2\operatorname{Re} \left( D_{\mathrm g}D_{\mathrm d}^* \right) \bigg].$$

The density cross term involves the real part of the coefficient product; the current involves its imaginary part. In particular, a relative phase of $\pi/2$ can remove the density cross term while producing nonzero current. Real exponential branches do not produce the oscillatory fringes of counterpropagating allowed waves.

What this means physically

• A real bound-state tail can be chosen real and has $j=0$.
• A pure decaying solution in a semi-infinite forbidden region has $j=0$.
• A stationary scattering state through a finite barrier has the same nonzero conserved current in allowed and forbidden regions.
• That under-barrier current is encoded by the complex mixture of growing and decaying local branches, not by assigning a real classical velocity to either branch.

Probability Current derives the general conservation law. The broader physical meaning of tunneling is developed in Quantum Tunneling.

Exponential Suppression Across an Interval

For $a<b$ inside one forbidden region, define the forbidden action

$$K(a,b;E) = \int_a^b \kappa(x;E)\,dx.$$

A branch that decays toward increasing $x$ has the local amplitude ratio

$$\frac{\psi_{\mathrm d}(b)} {\psi_{\mathrm d}(a)} \approx \sqrt{ \frac{\kappa(a)}{\kappa(b)} } \exp\left[ - \frac{K(a,b;E)}{\hbar} \right].$$

Its density ratio is

$$\frac{\lvert\psi_{\mathrm d}(b)\rvert^2} {\lvert\psi_{\mathrm d}(a)\rvert^2} \approx \frac{\kappa(a)}{\kappa(b)} \exp\left[ - \frac{2K(a,b;E)}{\hbar} \right].$$

The distinction between amplitude and probability exponents is fundamental. Across an entire smooth barrier bounded by two turning points, the same action controls the leading transmission probability, but turning-point matching and flux normalization are needed before this local decay ratio becomes a scattering coefficient.

Why the Prefactor Is Not Optional

The factor $\kappa^{-1/2}$ changes slowly compared with the exponential, but it is part of leading WKB. It follows from the transport equation and is required for a controlled approximation.

For one branch,

$$\psi(x) = A(x)e^{\pm\eta(x)},$$

the leading transport equation gives

$$\frac{d}{dx} \left( \kappa A^2 \right) = 0,$$

and hence

$$A(x) \propto \frac{1}{\sqrt{\kappa(x)}}.$$

Unlike the allowed-region factor $p^{-1/2}$, this should not be interpreted as a classical dwell-time density: there is no real classical trajectory with energy $E$ in the forbidden interval. It is an analytic continuation of the semiclassical transport factor.

When $K/\hbar$ is large, the exponential often dominates an order-of-magnitude estimate. That does not make the prefactor mathematically absent. It means only that its relative influence on a logarithmic estimate is smaller.

Validity in a Forbidden Region

The forbidden-region equation is

$$\psi'' - \frac{\kappa^2}{\hbar^2}\psi = 0.$$

The common slow-variation parameter is

$$\epsilon_{\mathrm f}(x) = \left\lvert \hbar \frac{\kappa'(x)}{\kappa^2(x)} \right\rvert \ll1.$$

Since

$$\kappa' = \frac{mV'}{\kappa},$$

one may also write

$$\epsilon_{\mathrm f} = \frac{ \hbar m\lvert V'\rvert }{ \kappa^3 }.$$

Direct residual

Substituting either branch $w_{\mathrm g}$ or $w_{\mathrm d}$ gives

$$w_\pm'' - \frac{\kappa^2}{\hbar^2}w_\pm ={} \biggl[ \frac34 \left( \frac{\kappa'}{\kappa} \right)^2 - \frac12 \frac{\kappa''}{\kappa} \biggr]w_\pm.$$

A dimensionless residual is

$$\mathcal R_{\mathrm f} = \frac{\hbar^2}{\kappa^2} \left[ \frac34 \left( \frac{\kappa'}{\kappa} \right)^2 - \frac12 \frac{\kappa''}{\kappa} \right].$$

A small $\lvert\mathcal R_{\mathrm f}\rvert$ supplements the first-derivative test when curvature or several length scales matter. It controls the local residual, not accumulated relative error in an exponentially small transmitted wave.

Turning-point layer

At a simple turning point $x_t$ with forbidden side $x\gt x_t$, let

$$V(x)-E \approx F(x-x_t), \qquad F\gt0.$$

Then

$$\kappa(x) = \sqrt{2mF(x-x_t)}$$

and

$$\begin{aligned} \epsilon_{\mathrm f} &= \frac12 \left( \frac{\ell_F}{x-x_t} \right)^{3/2}, \\ \ell_F &= \left( \frac{\hbar^2}{2mF} \right)^{1/3}. \end{aligned}$$

Leading WKB requires $x-x_t\gg\ell_F$. The apparent divergence $\kappa^{-1/2}$ at $x_t$ is a breakdown of the outer approximation, not a divergence of the exact wavefunction.

Abrupt interfaces

Inside a constant forbidden plateau, the exponential solutions are exact. At an abrupt edge of that plateau, however, the slow-variation condition is not defined. Exact continuity matching, not a smooth turning-point formula, handles the interface. A rectangular barrier can therefore be an exact benchmark for the interior exponent while still lying outside smooth WKB at its boundaries.

Worked Examples

Constant forbidden plateau

Let $V(x)=V_0$ with $V_0\gt E$. Then

$$\kappa_0 = \sqrt{2m(V_0-E)}$$

is constant, and

$$\psi(x) ={} \frac{D_{\mathrm g}}{\sqrt{\kappa_0}} e^{+\kappa_0(x-x_*)/\hbar} + \frac{D_{\mathrm d}}{\sqrt{\kappa_0}} e^{-\kappa_0(x-x_*)/\hbar}.$$

Because $\kappa_0'=\kappa_0''=0$, the WKB residual vanishes and these are exact interior solutions. For a finite rectangular barrier, the coefficients still require exact matching at both discontinuous interfaces; the local exactness of the interior basis does not make the smooth-turning-point machinery applicable there.

Linear forbidden ramp

For

$$V(x)-E = F(x-x_t), \qquad x\gt x_t,$$

the forbidden action from the turning point is

$$\begin{aligned} \eta(x) &= \frac{1}{\hbar} \int_{x_t}^{x} \sqrt{2mF(s-x_t)}\,ds \\ &= \frac{ 2\sqrt{2mF} }{ 3\hbar } (x-x_t)^{3/2}. \end{aligned}$$

The decaying outer solution is

$$\psi_{\mathrm d}(x) \propto \frac{ \exp\left[ - \dfrac{ 2\sqrt{2mF} }{ 3\hbar } (x-x_t)^{3/2} \right] }{ \bigl[2mF(x-x_t)\bigr]^{1/4} }.$$

This is the large-positive-argument asymptotic form of the decaying Airy solution, up to an overall constant. It is accurate only when $x-x_t\gg\ell_F$; the Airy function itself remains uniform through the turning point.

Harmonic-oscillator tail

For the harmonic oscillator,

$$V(x) = \frac12m\omega^2x^2,$$

write

$$E = \frac12m\omega^2a^2.$$

On the right forbidden tail $x\gt a$,

$$\kappa(x) = m\omega\sqrt{x^2-a^2}.$$

The forbidden action from the turning point is

$$K(a,x) ={} \frac{m\omega}{2} \bigg[ x\sqrt{x^2-a^2} - a^2 \operatorname{arcosh} \left( \frac{x}{a} \right) \bigg].$$

The decaying WKB tail is

$$\psi_{\mathrm d}(x) \propto \frac{ e^{-K(a,x)/\hbar} }{ \sqrt{m\omega} \left(x^2-a^2\right)^{1/4} }.$$

For $x\gg a$,

$$\frac{K(a,x)}{\hbar} \approx{} \frac{m\omega x^2}{2\hbar} - \frac{E}{\hbar\omega} \log\left(\frac{2x}{a}\right) - \frac{E}{2\hbar\omega}.$$

For the exact oscillator energy

$$E_n = \hbar\omega \left( n+\frac12 \right),$$

the exponential contributes a power $x^{n+1/2}$ multiplying the Gaussian, while the prefactor contributes $x^{-1/2}$. Therefore

$$\psi_{\mathrm d}(x) \propto x^n \exp\left[ - \frac{m\omega x^2}{2\hbar} \right]$$

at leading large-$x$ order. This matches the exact Hermite-function tail. WKB fails near $x=a$, but far into the forbidden region it recovers both the Gaussian exponent and the polynomial power.

The exact states are developed in Quantum Harmonic Oscillator.

Exercises

1. Let $D_{\mathrm g}=1$ and $D_{\mathrm d}=-i$. Find the current and the density cross term. Reverse the direction in which the forbidden action is measured and check the current sign.

Solution

The product $D_{\mathrm g}^*D_{\mathrm d}=-i$ gives $j=2/m>0$, while its real part vanishes, so there is no density cross term. Defining $\widetilde\eta=-\eta$ swaps the basis functions and coefficients. The Wronskian changes sign to $+2/\hbar$, and the coefficient product becomes $i$. Their combination still gives $j=2/m$: relabeling the basis cannot reverse physical flux.

2. A numerical decaying tail contains a relative growing-branch contamination $\varepsilon$ at action reference $\eta=0$. How large is the relative contamination after an increase $\Delta\eta$? What numerical strategy avoids amplifying it?

Solution

The unwanted-to-wanted ratio becomes $\varepsilon e^{2\Delta\eta}$. Forward integration deep into a forbidden tail is therefore unstable for computing the decaying solution. Integrating inward from a controlled asymptotic boundary, or using a suitably stabilized logarithmic-derivative/matching method, avoids following the unstable direction. Convergence in the endpoint location and integration tolerance must still be checked.

References

• C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I (Springer, 1999). Liouville–Green expansions, residuals, and turning-point asymptotics.
• M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics”, Reports on Progress in Physics 35, 315–397 (1972). Phase-integral branches, barriers, and semiclassical interpretation.
• J. Heading, An Introduction to Phase-Integral Methods (Methuen, 1962). Exponential bases, Stokes structure, and connection theory.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed. (Butterworth-Heinemann, 1981). Forbidden-region WKB, connection formulas, and barrier penetration.
• F. W. J. Olver, Asymptotics and Special Functions (A K Peters, 1997). Error-controlled exponential and Airy asymptotics.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed. (Springer, 1994). Standard WKB tunneling and bound-state examples.