WKB in Classically Allowed Regions

In a classically allowed region, the energy lies above the potential and the local classical momentum is real. Leading WKB then represents the wavefunction as a coherent sum of right- and left-moving oscillatory branches. Each branch carries constant probability current, while its density grows where the corresponding classical motion slows.

Required background. WKB approximation derives the phase and transport equations. Probability current supplies the conserved flux used to distinguish the two branches.

The Local Oscillatory Basis

Consider the stationary one-dimensional Schrödinger equation

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

with real $V(x)$. On an open interval $I$ where

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

define the positive local momentum and wavenumber

$$\begin{aligned} p(x) &= \sqrt{2m\bigl(E-V(x)\bigr)}, \\ k(x) &= \frac{p(x)}{\hbar}. \end{aligned}$$

Choose an arbitrary reference point $x_*\in I$ and define the accumulated phase

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

The leading allowed-region WKB basis is

$$u_\pm(x) = \frac{1}{\sqrt{p(x)}} \exp\left[ \pm i\theta(x;x_*) \right].$$

Thus the general local solution is

$$\psi_{\mathrm{WKB}}(x) \approx{} C_+u_+(x) + C_-u_-(x).$$

The $+$ branch has phase gradient $+p$ and the $-$ branch has phase gradient $-p$. For the Hamiltonian $H=p^2/(2m)+V$, their ray velocities are therefore

$$v_\pm(x) = \pm\frac{p(x)}{m}.$$

This direction assignment follows from current, not from visual inspection of a real-valued snapshot. The local action–phase dictionary is developed in Classical Action and Quantum Phase.

The reference point is a convention

Changing $x_*$ changes $\theta$ by a constant. If

$$\theta(x;\widetilde x_*) = \theta(x;x_*)-\alpha,$$

then the same wavefunction is obtained by

$$\widetilde C_+ = C_+e^{i\alpha}, \qquad \widetilde C_- = C_-e^{-i\alpha}.$$

Only relative phases fixed by boundary data or matching are observable. A lower limit written without explanation is not a physical origin of action.

Probability Current and Branch Direction

For a real scalar potential, the one-dimensional current is

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

Applied to the WKB basis,

$$\frac{u_\pm'}{u_\pm} = - \frac{p'}{2p} \pm i\frac{p}{\hbar}.$$

Since $\lvert u_\pm\rvert^2=1/p$, each basis function carries

$$j[u_\pm] = \pm\frac{1}{m}.$$

For the two-branch superposition with constant coefficients,

$$j_{\mathrm{WKB}} = \frac{ \lvert C_+\rvert^2 - \lvert C_-\rvert^2 }{m}.$$

The interference terms cancel from the current. They do not cancel from the density.

Flux-normalized branches

It is often convenient in scattering calculations to absorb $\sqrt m$ into the basis:

$$\chi_\pm(x) = \sqrt{\frac{m}{p(x)}} \exp\left[ \pm i\theta(x;x_*) \right].$$

Then

$$j[\chi_\pm] = \pm1.$$

These are unit-flux rather than unit-norm states. Stationary scattering states are generally not square-integrable, so flux normalization is the useful convention. Whenever amplitudes from different asymptotic regions are compared, current ratios, not coefficient magnitudes alone, determine reflection and transmission probabilities.

Traveling waves and standing waves

The density of the general WKB combination is

$$\lvert\psi_{\mathrm{WKB}}\rvert^2 ={} \frac{1}{p} \left( \lvert C_+\rvert^2 + \lvert C_-\rvert^2 \right) + \frac{2}{p} \operatorname{Re} \left[ C_+C_-^* e^{2i\theta} \right].$$

A pure branch has nonzero current and a smooth density envelope. If $\lvert C_+\rvert=\lvert C_-\rvert$, the net current vanishes and the wavefunction can be written, up to a constant overall phase, as

$$\psi_{\mathrm{stand}}(x) = \frac{B}{\sqrt{p(x)}} \cos\bigl(\theta(x)+\delta\bigr).$$

This is a standing wave built from two opposite fluxes. Zero net current does not mean that each branch has zero momentum; equal counterpropagating currents cancel. The phase gradient of the total standing wave is instead zero between its nodes, so it must not be identified with either branch action.

Why the Amplitude Scales as Inverse Square Root of Momentum

For one traveling branch, write

$$\psi(x) = A(x)e^{\pm i\theta(x)}.$$

Its leading current is

$$j \approx \pm \frac{p}{m} \lvert A\rvert^2.$$

Stationarity requires $dj/dx=0$, so

$$\frac{d}{dx} \left( p\lvert A\rvert^2 \right) = 0.$$

Therefore

$$\lvert A(x)\rvert \propto \frac{1}{\sqrt{p(x)}}.$$

The transport factor therefore expresses flux conservation:

$$\begin{aligned} \rho_\pm(x) &= \frac{\lvert C_\pm\rvert^2}{p(x)}, \\ j_\pm &= \rho_\pm v_\pm = \pm\frac{\lvert C_\pm\rvert^2}{m}. \end{aligned}$$

Where $p$ is smaller, the ray velocity is smaller and a fixed flux requires a larger probability density.

Across a smooth allowed region, decreasing $p(x)$ increases both the local de Broglie wavelength and the envelope $p(x)^{-1/2}$. The amplitude growth preserves current; it is not an independent probability source.

Classical dwell-time density

The same inverse-velocity weighting appears in classical mechanics. For periodic motion between two turning points, a classical trajectory crosses an interior point twice per period $T$. Its normalized dwell-time density is

$$P_{\mathrm{cl}}(x) = \frac{2}{T\lvert v(x)\rvert} = \frac{2m}{T p(x)}.$$

For a standing WKB wave,

$$\lvert\psi_{\mathrm{stand}}\rvert^2 = \frac{\lvert B\rvert^2}{p} \cos^2(\theta+\delta).$$

Coarse-graining over several local oscillations uses

$$\overline{\cos^2(\theta+\delta)} = \frac12,$$

and therefore

$$\overline{\lvert\psi_{\mathrm{stand}}\rvert^2} = \frac{\lvert B\rvert^2}{2p(x)}.$$

After normalization, this agrees with the classical dwell-time density in the semiclassical regime. The agreement concerns a locally averaged density. The exact quantum density retains nodes and interference fringes.

Validity Inside an Allowed Region

The condition $E\gt V$ makes the WKB solution oscillatory, but it does not by itself make WKB accurate. The local momentum must also vary slowly on the scale of a wavelength.

The common first-derivative test

The standard dimensionless parameter is

$$\epsilon_1(x) = \left\lvert \hbar\frac{p'(x)}{p^2(x)} \right\rvert \ll1.$$

Since

$$p' = - \frac{mV'}{p},$$

this can be written

$$\epsilon_1 = \frac{ \hbar m\lvert V'\rvert }{ p^3 }.$$

Equivalently, the fractional change in $p$ across a reduced wavelength $\hbar/p$ must be small. Numerical factors of $2\pi$ differ depending on whether one speaks of the reduced wavelength or the full de Broglie wavelength; the asymptotic requirement is the same.

A residual test that sees curvature

The stationary equation can be written

$$\psi'' + \frac{p^2}{\hbar^2}\psi = 0.$$

Substituting either leading branch $u_\pm$ gives

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

A direct dimensionless residual is therefore

$$\mathcal R(x) = \frac{\hbar^2}{p^2} \left[ \frac34 \left( \frac{p'}{p} \right)^2 - \frac12 \frac{p''}{p} \right].$$

A small $\lvert\mathcal R\rvert$ tests the local differential-equation residual against its kinetic scale. It is not by itself a bound on the solution error. This exposes a limitation of checking only $\epsilon_1$: at a point where $p'=0$, the first-derivative test vanishes even though $p''$ may still generate an error.

One can track the two derivative scales separately:

$$\epsilon_1 = \frac{\hbar\lvert p'\rvert}{p^2}, \qquad \epsilon_2 = \frac{\hbar^2\lvert p''\rvert}{p^3}.$$

For a generic smooth profile with one variation length, $\epsilon_2$ is of the same asymptotic order as $\epsilon_1^2$. Near special points or on multiscale profiles, it need not be.

Local accuracy is not global phase accuracy

Even when the residual is small everywhere, a small local correction to the wavenumber can accumulate over many oscillations. A phase-sensitive calculation should therefore report both:

1. a local slow-variation or residual estimate;
2. an estimate of the integrated phase error over the interval used.

For a spectrum, resonance, or interferometer, an order-one phase error can matter even when the relative error in a large action is small. The general distinction between local control and accumulated observable error is developed in Small Parameters and Error Estimates.

Distance from turning points

At a simple turning point $x_t$,

$$p(x_t)=0,$$

so the amplitude and derivative tests fail. The leading allowed-region formula is valid only on subintervals that stay outside the local turning-point layer. Extending $p^{-1/2}$ to $x_t$ and interpreting its divergence physically is incorrect; the exact wavefunction remains finite in the usual simple-turning-point problem.

Worked Examples

Constant potential

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

$$p_0 = \sqrt{2m(E-V_0)}$$

is constant, so

$$\theta(x;x_*) = \frac{p_0(x-x_*)}{\hbar}.$$

The WKB branches are plane waves,

$$u_\pm(x) = \frac{1}{\sqrt{p_0}} e^{\pm ip_0(x-x_*)/\hbar},$$

and they solve the Schrödinger equation exactly. Here $p'=p''=0$, so the residual vanishes. This benchmark checks branch signs, current normalization, and phase conventions.

Linear ramp away from its turning point

Take

$$V(x) = V_0+Fx, \qquad F\gt0,$$

and define

$$x_t = \frac{E-V_0}{F}.$$

The region $x\lt x_t$ is allowed. Its momentum is

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

Using the turning point only as a phase reference,

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

Away from $x_t$, the oscillatory basis is

$$u_\pm(x) = \frac{ e^{\mp i\zeta(x)} }{ \bigl[2mF(x_t-x)\bigr]^{1/4} }.$$

The first-derivative parameter is

$$\epsilon_1(x) = \frac{\hbar}{ 2\sqrt{2mF} (x_t-x)^{3/2} }.$$

Introduce the Airy length

$$\ell_F = \left( \frac{\hbar^2}{2mF} \right)^{1/3}.$$

Then

$$\epsilon_1 = \frac12 \left( \frac{\ell_F}{x_t-x} \right)^{3/2}.$$

The allowed-region WKB formula is controlled for $x_t-x\gg\ell_F$. Inside a layer of order $\ell_F$, the exact Airy solution, not the isolated WKB branches, supplies the uniform description.

Harmonic oscillator interior

For

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

write the energy as

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

The allowed interval is $\lvert x\rvert\lt a$, with

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

Taking the origin as phase reference gives

$$\int_0^x p(s)\,ds ={} \frac{m\omega}{2} \bigg[ x\sqrt{a^2-x^2} + a^2\arcsin\left(\frac{x}{a}\right) \bigg].$$

The amplitude envelope is

$$\frac{1}{\sqrt{p(x)}} = \frac{1}{ \sqrt{m\omega} \left(a^2-x^2\right)^{1/4} }.$$

It grows toward $\pm a$, warning that the interior approximation is approaching its turning-point failure. The first-derivative parameter is

$$\epsilon_1(x) = \frac{ \hbar\lvert x\rvert }{ m\omega \left(a^2-x^2\right)^{3/2} }.$$

At $x=0$, this parameter vanishes by symmetry, but WKB is not thereby exact. The curvature term in $\mathcal R$ remains, and the relevant interior action scale is

$$p(0)a = m\omega a^2 = \frac{2E}{\omega}.$$

The interior becomes semiclassical when $E/(\hbar\omega)$ is large. After oscillation averaging and normalization, the WKB standing-wave density tends to

$$\overline{\lvert\psi(x)\rvert^2} \approx \frac{1}{ \pi\sqrt{a^2-x^2} },$$

which is exactly the normalized classical dwell-time density away from the turning-point layers. The exact model is developed in Quantum Harmonic Oscillator; its global WKB spectrum and turning-point phases are standard applications of Bohr–Sommerfeld Quantization.

High-energy phase through a smooth weak potential

Suppose $\lvert V(x)\rvert/E$ is small throughout an allowed interval and define

$$p_0 = \sqrt{2mE}.$$

Expanding the local momentum gives

$$\begin{aligned} p(x) &= p_0 \sqrt{1-\frac{V(x)}{E}} \\ &\approx p_0 - \frac{m}{p_0}V(x). \end{aligned}$$

The accumulated phase is therefore

$$\theta(x;x_*) \approx \frac{p_0(x-x_*)}{\hbar} - \frac{m}{\hbar p_0} \int_{x_*}^{x} V(s)\,ds.$$

The first term is free propagation; the second is the leading phase delay or advance. The amplitude changes more mildly:

$$\frac{1}{\sqrt{p(x)}} \approx \frac{1}{\sqrt{p_0}} \left[ 1+ \frac{V(x)}{4E} \right].$$

This local single-branch approximation does not by itself compute the reflected amplitude. Reflection is branch conversion determined by global variation, boundaries, or complex turning-point structure.

Exercises

1. A unit-flux traveling WKB branch enters a region where its positive momentum falls from $p_a$ to $p_a/4$. Find its amplitude, density, speed, and wavelength ratios.

Solution

The amplitude doubles, density quadruples, speed falls by four, and wavelength grows by four. Thus $\rho v$ remains constant. A pure branch need not be real: plotting its real part produces oscillations, but its modulus has no fringes.

2. For the oscillator interior, evaluate $\mathcal R$ at $x=0$. Does the exactness of the WKB energy spectrum imply exact WKB wavefunctions there?

Solution

Here $p'=0$, $p''/p=-1/a^2$, and $p(0)=m\omega a$, giving

$$\mathcal R(0)=\frac{\hbar^2}{2m^2\omega^2a^4} =\frac18\left(\frac{\hbar\omega}{E}\right)^2.$$

It equals $1/2$ for the ground-state energy and tends to zero at high energy. Correct energy levels do not make the leading wavefunction exact.

References

• C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I (Springer, 1999). Liouville–Green asymptotics, error structure, and connection methods.
• M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics”, Reports on Progress in Physics 35, 315–397 (1972). Phase-integral interpretation, branch structure, and semiclassical validity.
• J. Heading, An Introduction to Phase-Integral Methods (Methuen, 1962). Traveling-wave bases, phase integrals, and connection theory.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed. (Butterworth-Heinemann, 1981). WKB waves, current interpretation, turning points, and quantization.
• F. W. J. Olver, Asymptotics and Special Functions (A K Peters, 1997). Error-controlled Liouville–Green approximations and uniform asymptotics.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed. (Springer, 1994). WKB wavefunctions, bound-state interpretation, and standard examples.