WKB and Semiclassical Methods

Semiclassical methods use classical motion to organize quantum phases and amplitudes. They can predict approximate energy levels, tunneling probabilities, and interference patterns when a relevant action scale is large compared with $\hbar$. They still describe quantum waves: several classical paths can interfere, and forbidden regions can carry exponentially small amplitudes.

The most direct entry is one-dimensional WKB. Learn its oscillating and exponential solutions first, then how turning-point matching joins them. Stationary phase and trajectory-based propagation provide a second route through the chapter.

Helpful background. The semiclassical overview explains the physical limit; the stationary Schrödinger equation supplies the wave equation. Asymptotic analysis and Hamilton–Jacobi theory explain expansions and action phases.

A slowly changing wavelength

Consider a particle of constant mass $m$ in a smooth, real, one-dimensional potential. Where $E>V(x)$, define the positive momentum magnitude

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

A right-moving WKB branch has the form

$$\psi_+(x)\approx\frac{C}{\sqrt{p(x)}} \exp\!\left[\frac{i}{\hbar}\int^x p(x')\,dx'\right].$$

The phase advances more rapidly where the momentum is larger. The amplitude decreases there, so the probability flux stays constant: faster motion requires less density for the same flow. Classical action and quantum phase explains this phase–momentum relation; the WKB approximation derives the amplitude and its corrections.

A useful local diagnostic is

$$\epsilon_{\mathrm{WKB}}(x) =\frac{\hbar|p'(x)|}{p(x)^2}\ll1.$$

It asks whether the momentum changes little over a reduced wavelength $\hbar/p$. It is a diagnostic within a smooth asymptotic regime, not a stand-alone error theorem for every rapidly structured potential. Large orbit action does not ensure local validity everywhere, and a numerical value of a dimensionful $\hbar$ says nothing by itself.

What happens at a turning point

At a turning point $x_t$, $E=V(x_t)$ and the WKB prefactor diverges. The exact wavefunction need not diverge. Rather, the oscillatory and exponential descriptions must be joined through a different local approximation.

For a simple turning point, $V'(x_t)\ne0$, the linearized potential gives an Airy equation on the length scale

$$\ell_t=\left(\frac{\hbar^2}{2m|V'(x_t)|}\right)^{1/3}.$$

This solution remains finite through the turn and matches WKB away from it. It also fixes relative amplitudes and phase shifts that cannot be guessed from the exponent alone.

Follow these three pages together: allowed regions for traveling and standing waves, forbidden regions for growing and decaying branches, and turning-point connection formulas for their matching. An abrupt step, hard wall, or pair of merging turning points needs its own local treatment; the simple Airy formula does not cover all boundaries.

From local waves to measurable quantities

Boundary conditions turn the local solutions into predictions. For a smooth confining well with two simple turning points, leading Bohr–Sommerfeld quantization gives

$$\int_{x_1(E_n)}^{x_2(E_n)}p(x;E_n)\,dx \approx\pi\hbar\left(n+\frac12\right).$$

The half-integer shift comes from the two smooth turns. For the harmonic oscillator this leading rule happens to give the exact energies; generic wells receive corrections. Hard-wall quantization has different reflection phases.

For a smooth barrier, use the positive forbidden momentum magnitude $\kappa(x)=\sqrt{2m[V(x)-E]}$. Here $\kappa$ has momentum units, not inverse-length units. The transmission probability for a thick barrier has the leading exponential dependence

$$T(E)\sim\mathcal P(E)e^{-2K(E)/\hbar}, \qquad K(E)=\int_{x_1}^{x_2}\kappa(x)\,dx.$$

This requires $K/\hbar\gg1$ together with suitable local matching. The prefactor $\mathcal P$ and flux normalization still matter; near the barrier top the two-turning-point approximation changes. Barrier penetration explains the calculation, and the Gamow factor applies it to a repulsive Coulomb barrier.

More dimensions and moving wave packets

In an oscillatory integral, rapidly changing phases tend to cancel, while neighborhoods of stationary phase can contribute coherently. Stationary phase in quantum mechanics turns this idea into calculations. For propagation, the stationary paths are classical trajectories, but their contributions are complex amplitudes that must be added before taking a probability.

What you want to understand	Continue with
Waves following families of rays	Multidimensional WKB
Motion represented by classical paths between endpoints	Semiclassical propagator
How trajectory focusing changes the amplitude	Van Vleck determinant
Phase changes when rays turn or focus	Maslov index
Quantization on invariant tori in integrable systems	EBK quantization
Localized packets and phase-space trajectories	Coherent-state semiclassics
Which observables and time intervals admit a classical comparison	Semiclassical limits and correspondence

At a caustic, nearby rays focus and an isolated-ray amplitude becomes singular. Combining the nearby contributions in a uniform approximation replaces that singular expression. Long propagation times introduce a separate difficulty: small action errors can accumulate into large phase errors. The relevant phase difference is $\delta S/\hbar$, so a small relative action error need not preserve an interference pattern.

Stationary phase alone also does not explain the disappearance of interference. The classical limit separates semiclassical propagation, state preparation, coarse resolution, and environmental decoherence.

Exercises

Check the flux. For the right-moving WKB branch above, evaluate $j=(\hbar/m)\operatorname{Im}(\psi^*\psi')$ with real positive $p(x)$. What would go wrong if the amplitude were constant?

Solution

The derivative of $p^{-1/2}$ contributes a real term to $\psi^*\psi'$ and therefore no current. The phase derivative contributes $i|C|^2/\hbar$, giving $j=|C|^2/m$. This identity is exact for the displayed approximate wavefunction. A constant amplitude would instead give $j\propto p(x)$, which varies with position and violates stationary current conservation.

A phase-accuracy target. A relevant action difference satisfies $|\Delta S|/\hbar=10^4$. How accurately must it be computed to keep its phase error below $0.1$ radians?

Solution

The phase error is $|\delta(\Delta S)|/\hbar$. Thus the relative error must obey $|\delta(\Delta S)/\Delta S|<10^{-5}$. The action difference, not an arbitrary additive constant in either action, determines the observable interference phase.

A hard-wall check. For an infinite well of width $L$, the momentum is constant inside and the exact condition is $pL=n\pi\hbar$, $n=1,2,\ldots$. Why is the smooth-well half-integer rule inappropriate?

Solution

The walls enforce nodes directly. They are not simple turning points with a finite nonzero potential slope, so the Airy connection phase does not apply. Matching the two node conditions gives the integer rule. WKB’s local plane-wave form is exact inside the well, but the boundary phase must still be supplied correctly.

References

• Bender, C. M., and S. A. Orszag. Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory. Springer (1999). WKB, turning points, and uniform approximations.
• Berry, M. V., and K. E. Mount. “Semiclassical approximations in wave mechanics.” Reports on Progress in Physics 35, 315–397 (1972). doi:10.1088/0034-4885/35/1/306. Phases, amplitudes, and multidimensional semiclassics.
• Zwiebach, B. “Chapter 3: Semiclassical Approximation.” Quantum Physics III, MIT OpenCourseWare (2018). An introduction to one-dimensional WKB and its physical interpretation.