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.

For the chapter-wide map from local WKB branches to turning-point repair, quantization, tunneling, and trajectory-based propagators, begin with WKB and Semiclassical Methods.

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.

Ansatz

Use a local phase form

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

where $S(x)$ varies rapidly compared with the amplitude $A(x)$. 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 approximation is semiclassical because the phase appears as $S/\hbar$. At leading order, this phase solves the Hamilton–Jacobi equation. Hamilton–Jacobi Theory Preview shows how this equation emerges from the exact wavefunction phase equation and why the amplitude correction cannot always be neglected. When $S$ is large compared with $\hbar$, small changes in action produce large phase changes. The broader classical-limit idea is introduced in Correspondence Principle and mapped in Classical Limit; the focused bridge is Semiclassical Limit Overview, and the Toolkit mechanism is Semiclassical Limit.

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

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

The factor $1/\sqrt{p(x)}$ has a simple physical meaning: in regions where the classical particle moves slowly, probability density is enhanced.

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

The neglected term is small when the local wavelength changes slowly. A common form 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.

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.

The leading WKB form is often excellent for phases and exponential factors but less reliable for prefactors. When the exponent is large, the exponential dependence dominates; when the exponent is modest, connection formulas and higher-order corrections matter.

Common Mistakes

• Applying the leading WKB form directly at $p(x)=0$.
• Forgetting the amplitude factor $1/\sqrt{p(x)}$.
• Treating the exponent in the wavefunction as the exponent in probability. Probabilities involve squared amplitudes.
• Assuming WKB requires $\hbar$ literally small. The real requirement is that the dimensionless action scale be large compared with $\hbar$.
• Using WKB for abrupt discontinuities. Piecewise-constant potentials are usually better handled by exact matching.

Exercises

1. Starting from the WKB ansatz, derive the transport equation $2A'p+Ap'=0$ in an allowed region.

Solution

Substitution gives

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

At leading order $S'=\pm p$. At order $\hbar$, the coefficient of $i\hbar$ must vanish:

$$2\frac{A'}{A}S' + S'' = 0.$$

Using $S'=\pm p$ and $S''=\pm p'$ gives

$$2\frac{A'}{A}p+p'=0,$$

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

2. Show that $A(x)\propto 1/\sqrt{p(x)}$ solves the transport equation.

Solution

The transport equation can be written

$$\frac{A'}{A} = - \frac12 \frac{p'}{p}.$$

Integrating gives

$$\log A = - \frac12\log p + \text{constant},$$

so $A\propto p^{-1/2}$.

3. Explain why leading WKB is not appropriate for a rectangular barrier interface.

Solution

At a rectangular interface the potential changes discontinuously, so the local wavelength changes abruptly rather than slowly. WKB assumes slow variation over a local wavelength. The rectangular barrier is better solved by exact continuity matching, while WKB is designed for smooth barriers.

References

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