WKB Approximation

The WKB approximation is the leading semiclassical approximation for a wavefunction in a slowly varying potential. It assumes that the local de Broglie wavelength changes only slightly over one wavelength, so the wavefunction looks locally like a plane wave with slowly varying amplitude and phase.

Required background. The stationary Schrödinger equation and boundary conditions determine the differential equation and the physical combination of its solutions.

The local meaning of $S/\hbar$, including its momentum, energy, wavefront, interference, and gauge-covariance interpretations, is developed in Classical Action and Quantum Phase.

For a one-dimensional stationary state,

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

define the local classical momentum in an allowed region by

$$p(x) = \sqrt{2m\left(E-V(x)\right)}.$$

The WKB approximation gives oscillatory wavefunctions where $E\gt V(x)$ and exponential wavefunctions where $E\lt V(x)$. It fails at turning points, where $E=V(x)$ and $p(x)=0$; the repair uses local Airy Functions and is the subject of Turning Points and Connection Formulas.

Local action and amplitude

Use a local phase form

$$\psi(x) = A(x) \exp\left( \frac{i}{\hbar}S(x) \right),$$

where the dimensionless phase $S/\hbar$ changes by order one over a local wavelength while $A$ changes by a small fraction. Substituting into the stationary Schrödinger equation gives

$$\left(S'\right)^2 - i\hbar \left( 2\frac{A'}{A}S' + S'' \right) - \hbar^2\frac{A''}{A} = 2m\left(E-V\right).$$

WKB organizes this equation as an expansion in powers of $\hbar$ relative to the action scale set by the problem.

The general language of such limiting expansions is summarized in Asymptotic Analysis.

Leading Phase

At leading order,

$$\left(S'\right)^2 = p^2(x),$$

so

$$S'(x)=\pm p(x).$$

Thus the rapidly varying phase is the classical action integral:

$$S_\pm(x) = \pm \int^x p(x')\,dx'.$$

The phase solves the leading Hamilton–Jacobi equation. Its derivative, rather than the arbitrary additive constant in $S$, determines the local momentum.

Amplitude

The next order gives the transport equation

$$2A'p+Ap'=0.$$

Equivalently,

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

Therefore

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

In a classically allowed region, the leading WKB wavefunction is

$$\psi(x) \approx \frac{C_+}{\sqrt{p(x)}} \exp\left( \frac{i}{\hbar}\int^x p(x')\,dx' \right) + \frac{C_-}{\sqrt{p(x)}} \exp\left( - \frac{i}{\hbar}\int^x p(x')\,dx' \right).$$

For one traveling branch, the density is proportional to $1/p$: a conserved flux spends more time where motion is slow. For a standing wave this interpretation applies after averaging its rapid interference fringes, not to its pointwise nodal density.

WKB in Classically Allowed Regions develops this result into traveling and standing branches, current and unit-flux normalizations, dwell-time density, residual tests, and worked smooth-potential examples.

Forbidden Regions

In a classically forbidden region, define

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

The WKB solutions become

$$\psi(x) \approx \frac{D_+}{\sqrt{\kappa(x)}} \exp\left( \frac{1}{\hbar}\int^x \kappa(x')\,dx' \right) + \frac{D_-}{\sqrt{\kappa(x)}} \exp\left( - \frac{1}{\hbar}\int^x \kappa(x')\,dx' \right).$$

One branch grows and one branch decays. Physical boundary conditions usually select the decaying branch far inside a forbidden tail, but in tunneling problems both branches may be needed locally to match across two turning points.

WKB in Classically Forbidden Regions develops the orientation conventions, boundary-condition logic, under-barrier current, residual tests, and tail benchmarks for these exponential branches.

Validity Condition

A useful slow-variation diagnostic is

$$\left\lvert \hbar \frac{p'}{p^2} \right\rvert \ll 1.$$

Equivalently, if

$$\lambda_{\mathrm{dB}}(x) = \frac{2\pi\hbar}{p(x)},$$

then WKB requires the potential and wavelength to vary little over one local wavelength.

For forbidden regions the analogous condition is

$$\left\lvert \hbar \frac{\kappa'}{\kappa^2} \right\rvert \ll 1.$$

Both conditions fail near a simple turning point because $p$ or $\kappa$ goes to zero. The first derivative alone is not sufficient for an arbitrary smooth potential. Substituting $A=p^{-1/2}$ back into the equation leaves the dimensionless local residual

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

Requiring both $|\hbar p'/p^2|\ll1$ and $|\hbar^2p''/p^3|\ll1$ controls these local terms. Replace $p$ by $\kappa$ in a forbidden region. A small residual is a local consistency check; a global error estimate must also control accumulation over the interval and sensitivity to boundary conditions.

What WKB Does and Does Not Say

WKB is local. It gives approximate solutions in regions where the potential varies slowly, but a physical wavefunction also requires global boundary conditions and matching. For bound states, matching across two turning points produces quantization. For barriers, matching across two turning points produces the tunneling exponent.

For an opaque smooth barrier, the leading action often determines the dominant logarithm of transmission. A small relative error in that action can still produce a substantial relative error in an exponentially small probability. Connection formulas determine leading prefactors, and higher orders are needed when that accuracy matters.

Exercises

1. Let $p(x)=p_0[1+\eta\cos(x/L)]$, with $0<\eta<1$. At $x=0$, does $p'=0$ guarantee a small WKB residual?

Solution

No. At that point $p''=-p_0\eta/L^2$, so

$$\epsilon_R(0)=\frac{\eta}{2(1+\eta)^3} \left(\frac{\hbar}{p_0L}\right)^2.$$

It can be large even though the first-derivative diagnostic vanishes. A rapidly varying, small-amplitude modulation can expose this failure.

2. Why is WKB exact inside each constant-potential region of a rectangular barrier but insufficient at the interfaces?

Solution

Constant $p$ or $\kappa$ gives exact plane waves or exponentials and zero local residual. At a discontinuity the slow-variation assumption fails. Continuity of $\psi$ and $\psi'$ supplies the interface coefficients; a smooth Airy turning-point formula does not apply there.

References

• C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Springer, 1999.
• M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics”, Reports on Progress in Physics 35, 315-397, 1972.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Butterworth-Heinemann, 1981.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.