Multidimensional WKB Preview

Multidimensional WKB replaces the single position coordinate of ordinary WKB by a family of classical rays in configuration space. The rapidly varying phase solves a Hamilton–Jacobi, or eikonal, equation. The slowly varying amplitude is transported by conservation of probability flux through a ray tube.

That concise statement hides the new difficulty. In one dimension, a local momentum sign usually labels the two branches. In several dimensions, neighboring rays can focus, intersect in configuration space, and generate several action branches at the same point. At a caustic the ray projection loses rank and an isolated WKB amplitude diverges. Several distinct regular rays can also reach the same point without forming a caustic.

Required background. WKB approximation gives the one-dimensional expansion; Hamilton–Jacobi theory explains characteristics. We use a scalar wavefunction in flat Cartesian coordinates, with smooth real $V$, and consider branches away from sources and singular boundaries.

Stationary WKB Ansatz

Consider a scalar particle in $d$ Euclidean dimensions:

$$\left[ - \frac{\hbar^2}{2m}\nabla^2 + V(\mathbf q) \right] \psi(\mathbf q) = E\psi(\mathbf q).$$

In a region with a rapidly varying phase and a slowly varying envelope, write one local branch as

$$\psi(\mathbf q) = A(\mathbf q) \exp\left[ \frac{i}{\hbar} S(\mathbf q) \right].$$

For the leading allowed-region construction, $S$ and $A$ may be taken real. The phase function $S$ has units of action.

Define the eikonal residual

$$\mathcal E[S] = \frac{ \lvert\nabla S\rvert^2 }{2m} + V-E$$

and the transport expression

$$\mathcal T[A,S] = 2\nabla A\cdot\nabla S + A\nabla^2S.$$

Direct substitution into the Schrödinger equation gives

$$0 ={} A\mathcal E[S] - \frac{i\hbar}{2m} \mathcal T[A,S] - \frac{\hbar^2}{2m} \nabla^2A.$$

This equation is exact for the ansatz. The WKB approximation organizes its three terms by how many slow derivatives or explicit powers of $\hbar$ they contain.

Eikonal Equation

At leading order,

$$\mathcal E[S]=0,$$

or

$$\frac{ \lvert\nabla S\rvert^2 }{2m} + V(\mathbf q) = E.$$

Define the local branch momentum by

$$\mathbf p(\mathbf q) = \nabla S(\mathbf q).$$

Then

$$\lvert\mathbf p\rvert = \sqrt{ 2m \bigl(E-V(\mathbf q)\bigr) }.$$

This is the time-independent Hamilton–Jacobi equation. It determines the phase surfaces $S(\mathbf q)=\text{constant}$ and the momentum field normal to them.

The equation determines only the magnitude of $\nabla S$ directly. Boundary data select its direction and may generate several solutions. A point reached by several classical rays generally has several local action branches $S_\gamma$, not one globally single-valued phase.

Characteristics Are Classical Rays

The characteristics of the eikonal equation are classical trajectories at energy $E$. With

$$H(\mathbf q,\mathbf p) = \frac{\lvert\mathbf p\rvert^2}{2m} + V(\mathbf q),$$

they obey

$$\begin{aligned} \dot{\mathbf q} &= \frac{\partial H}{\partial\mathbf p} = \frac{\mathbf p}{m}, \\ \dot{\mathbf p} &= - \frac{\partial H}{\partial\mathbf q} = - \nabla V. \end{aligned}$$

Along a characteristic,

$$\mathbf p=\nabla S,$$

and the action changes according to

$$\frac{dS}{dt} = \mathbf p\cdot\dot{\mathbf q}.$$

Thus the eikonal partial differential equation can be solved locally by launching a family of classical rays from suitable boundary data and accumulating the action along them.

The phase surfaces are wavefronts. Rays point in the direction of $\nabla S$ and therefore cross the wavefronts orthogonally for the scalar Hamiltonian used here. In magnetic fields, anisotropic media, band Hamiltonians, or constrained coordinates, velocity need not be parallel to canonical momentum; the ray construction must then use the full Hamiltonian characteristics.

Transport Equation

At the next order, set

$$\mathcal T[A,S]=0.$$

The amplitude obeys

$$2\nabla S\cdot\nabla A + A\nabla^2S = 0.$$

Multiplying by $A$ gives the conservation form

$$\nabla\cdot \left( A^2\nabla S \right) = 0.$$

For a real leading amplitude, the probability current is

$$\mathbf j = \frac{\hbar}{m} \operatorname{Im} \left( \psi^*\nabla\psi \right) \approx \frac{A^2}{m} \nabla S.$$

Consequently,

$$\nabla\cdot\mathbf j \approx 0.$$

The amplitude is not a decorative normalization. It is the factor that makes the classical ray family carry conserved quantum probability flux.

Evolution along one ray

Let

$$\mathbf v = \frac{\nabla S}{m}.$$

The transport equation becomes

$$\mathbf v\cdot\nabla A + \frac{A}{2} \nabla\cdot\mathbf v = 0.$$

Since $d/dt=\mathbf v\cdot\nabla$ along a ray,

$$\frac{d\ln A}{dt} = - \frac12 \nabla\cdot\mathbf v.$$

An expanding ray family has $\nabla\cdot\mathbf v\gt0$ and a decreasing amplitude. A focusing family has $\nabla\cdot\mathbf v\lt0$ and an increasing amplitude.

Ray Tubes and Jacobians

Label rays by $\boldsymbol\alpha=(\alpha_1,\ldots,\alpha_{d-1})$ and use $t$ as a parameter along them:

$$\mathbf q = \mathbf q(t,\boldsymbol\alpha).$$

The configuration-space ray Jacobian is

$$J_{\mathbf q} = \left| \det \frac{ \partial\mathbf q }{ \partial(t,\boldsymbol\alpha) } \right|.$$

Before the projection becomes singular,

$$\frac{d\ln J_{\mathbf q}}{dt} = \nabla\cdot\mathbf v.$$

Combining this relation with amplitude transport gives

$$\frac{d}{dt} \left( A^2J_{\mathbf q} \right) = 0.$$

Hence

$$A(t,\boldsymbol\alpha) ={} A(t_0,\boldsymbol\alpha) \times \left[ \frac{ J_{\mathbf q}(t_0,\boldsymbol\alpha) }{ J_{\mathbf q}(t,\boldsymbol\alpha) } \right]^{1/2}.$$

Equivalently, for a thin ray tube crossing a transverse area element $d\Sigma$,

$$A^2v_\perp\,d\Sigma = \text{constant},$$

where $v_\perp$ is the velocity component through that cross section.

Left: transport conserves $A^2v_\perp d\Sigma$, so a narrowing ray tube increases the WKB amplitude. Right: at a fold caustic, the ray-label map has $J_{\mathbf q}=0$; two real branches merge and the isolated-branch amplitude diverges, while the exact wave is described by a uniform approximation.

The Jacobian here is a configuration-space projection Jacobian. Hamiltonian flow remains nonsingular and volume-preserving in full phase space. A caustic occurs because a smooth phase-space manifold projects singularly onto configuration space, not because classical phase-space evolution itself ceases to exist.

One-Dimensional Limit

In one dimension, a stationary branch carries

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

Constant flux therefore requires

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

The familiar WKB prefactor is thus the one-dimensional form of ray-tube transport. There is no transverse area, but the local speed still changes the amount of probability density needed to carry a fixed current.

Basic Examples

Plane wave

For $V=V_0$ and $E\gt V_0$, choose a constant momentum satisfying

$$\frac{\lvert\mathbf p_0\rvert^2}{2m} + V_0 = E.$$

Then

$$S(\mathbf q) = \mathbf p_0\cdot\mathbf q$$

has

$$\nabla^2S=0.$$

The transport equation permits constant $A$, and the WKB branch is an exact plane-wave solution.

Radial outgoing wave

For a free particle outside a point source, take

$$S(\mathbf q)=pr, \qquad r=\lvert\mathbf q\rvert.$$

Radial flux through a sphere in $d$ dimensions gives

$$A^2 \frac{p}{m} r^{d-1} = \text{constant}.$$

Thus

$$A(r) \propto r^{-(d-1)/2}.$$

In three dimensions,

$$\psi_{\mathrm{out}}(r) \propto \frac{1}{r} \exp\left( \frac{ipr}{\hbar} \right).$$

The inverse-distance amplitude is the geometric spreading of a ray bundle, not an extra assumption about spherical waves. More generally, $r^{-(d-1)/2}$ is the leading large-$pr/\hbar$ radial amplitude; subleading inverse-radius terms are retained by the exact Bessel or Hankel solution.

Several classical paths

An obstacle, lensing potential, curved boundary, or nontrivial classical motion can send several rays to the same point. Each ray carries its own action and transport amplitude. Define its total semiclassical phase by

$$\Phi_\gamma = \frac{S_\gamma}{\hbar} - \frac{\pi}{2}\mu_\gamma.$$

The local semiclassical field is then a coherent branch sum rather than a single eikonal:

$$\psi(\mathbf q) \approx \sum_\gamma A_\gamma(\mathbf q) e^{i\Phi_\gamma}.$$

The integer $\mu_\gamma$ records caustic and representation phase changes under a chosen Maslov-index convention.

For each unordered pair, define

$$\mathcal I_{\gamma\gamma'} = A_\gamma A_{\gamma'}^* e^{i(\Phi_\gamma-\Phi_{\gamma'})}.$$

The probability contains the pair-interference terms

$$\lvert\psi\rvert^2 \approx{} \sum_\gamma \lvert A_\gamma\rvert^2 + 2 \sum_{\gamma<\gamma'} \operatorname{Re} \mathcal I_{\gamma\gamma'}.$$

Classical rays organize the branches, but the branches still interfere quantum mechanically.

Caustics

Away from a caustic, a local action branch represents a Lagrangian manifold in phase space as the graph

$$\Lambda = \left\{ (\mathbf q,\mathbf p) : \mathbf p=\nabla S(\mathbf q) \right\}.$$

The projection

$$\pi_{\mathbf q}: \Lambda\longrightarrow \text{configuration space}$$

can lose differential rank when the smooth phase-space manifold is continued beyond this graph chart. At such a point,

$$J_{\mathbf q}=0,$$

and the transport formula predicts

$$A\propto J_{\mathbf q}^{-1/2}.$$

The divergence is not a physical blow-up of the exact wave. It says that one configuration-space phase branch is a bad local representation.

Near a generic fold caustic, two stationary branches coalesce and an Airy-type uniform approximation replaces their separate divergent terms. Cusp caustics require a different canonical integral, often of Pearcey type. In each case the local canonical integral replaces the separate branches:

do not evaluate an isolated-branch WKB formula at a point where its ray Jacobian vanishes.

Crossing a caustic also changes the phase associated with the square-root amplitude. Maslov Index explains the resulting phase correction.

Relation to Ray Optics

For a scalar Helmholtz field,

$$\left[ \nabla^2 + k_0^2n^2(\mathbf q) \right] u(\mathbf q) = 0,$$

the geometrical-optics ansatz is

$$u = a e^{ik_0\mathcal L}.$$

At leading order,

$$\lvert\nabla\mathcal L\rvert^2 = n^2,$$

and the optical amplitude obeys

$$\nabla\cdot \left( a^2\nabla\mathcal L \right) = 0.$$

The quantum stationary equation can likewise be written

$$\left[ \nabla^2 + \frac{p^2(\mathbf q)}{\hbar^2} \right] \psi = 0,$$

where

$$p^2(\mathbf q) = 2m \bigl(E-V(\mathbf q)\bigr).$$

The correspondence is:

• optical length $k_0\mathcal L$ corresponds to quantum phase $S/\hbar$;
• optical rays correspond to classical trajectories;
• optical intensity transport corresponds to probability-current transport;
• optical focal caustics correspond to singular configuration-space projections of classical trajectory families.

The analogy is structural, not complete. Electromagnetic waves have polarization, interfaces impose vector boundary conditions, and quantum mechanics allows classically forbidden propagation, spin transport, gauge phases, and operator-valued internal structure.

Time-Dependent Extension

For

$$\psi(\mathbf q,t) = A(\mathbf q,t) \exp\left[ \frac{i}{\hbar} S(\mathbf q,t) \right],$$

the leading phase satisfies

$$\frac{\partial S}{\partial t} + \frac{ \lvert\nabla S\rvert^2 }{2m} + V(\mathbf q,t) = 0.$$

The leading amplitude satisfies

$$\frac{\partial A}{\partial t} + \frac{\nabla S}{m}\cdot\nabla A + \frac{A}{2m}\nabla^2S = 0.$$

Writing $\rho=A^2$ gives

$$\frac{\partial\rho}{\partial t} + \nabla\cdot \left( \rho \frac{\nabla S}{m} \right) = 0.$$

This is the time-dependent ray-density transport law. The Semiclassical Propagator and Van Vleck Determinant reorganize the same geometry in terms of trajectories joining fixed endpoints.

Forbidden Regions in Several Dimensions

In a forbidden region, a decaying branch may be written

$$\psi(\mathbf q) \sim B(\mathbf q) \exp\left[ - \frac{1}{\hbar} W(\mathbf q) \right].$$

The leading real decay action obeys

$$\lvert\nabla W\rvert^2 = 2m \bigl(V(\mathbf q)-E\bigr).$$

This is an imaginary-momentum eikonal equation. In more than one dimension, the exponent is not generally obtained by choosing an arbitrary straight path and integrating

$$\sqrt{2m(V-E)}.$$

Boundary data and the eikonal equation determine the relevant decay surface and dominant paths. Several complex or Euclidean trajectories can compete, and Stokes switching can change which branches contribute. Detailed multidimensional tunneling is therefore naturally connected to instanton and complex-trajectory methods rather than to a naive line integral.

From Local Rays to Global Quantization

The eikonal and transport equations are local. A global wavefunction must also satisfy:

• boundary and matching conditions;
• single-valuedness or bundle-valued consistency;
• phase changes at caustics;
• topology of the underlying classical invariant set;
• interference among all relevant branches.

For a classically integrable system, invariant tori and their cycles provide the global structure. EBK Quantization imposes phase consistency on those cycles and includes the Maslov correction.

For nonintegrable systems, one generally cannot introduce one global action function or one set of action-angle variables covering the motion. Local WKB branches can still exist, but ray proliferation, caustics, and long-time instability make global assembly harder.

Validity Conditions

There is no single scalar test that covers every multidimensional failure. The following local checks address distinct failures.

Slow variation

With

$$p = \sqrt{2m(E-V)},$$

one expects a phase-gradient condition such as

$$\epsilon_p = \hbar \frac{\lvert\nabla p\rvert}{p^2} \ll1$$

and an amplitude residual condition such as

$$\epsilon_A = \frac{ \hbar^2 \lvert\nabla^2A\rvert }{ p^2\lvert A\rvert } \ll1.$$

These are local diagnostics, not universal error bounds.

Nondegenerate projection

The branch also requires

$$J_{\mathbf q}\ne0.$$

Slowly varying $p$ does not protect a ray family from a caustic.

Branch separation

If the neighborhoods contributing around stationary points overlap in the underlying integration variables, the isolated-saddle expansion fails and a uniform approximation must retain them together. A small difference between action values alone is not a failure criterion: distinct, well-separated paths can have equal actions and interfere normally.

Appropriate internal structure

For degenerate bands, spinors, molecules with coupled electronic states, or polarized waves, the amplitude may be vector- or matrix-valued. Berry connections and nonadiabatic couplings then enter the transport equation. The scalar formula on this page is insufficient.

Exercises

1. For a free radial branch in $d$ dimensions, transport gives $A=r^{-(d-1)/2}$. Compute $\nabla^2A/A$ away from $r=0$ and identify when the leading outgoing wave is exact.

Solution

Using $\nabla^2r^a=a(a+d-2)r^{a-2}$,

$$\frac{\nabla^2A}{A}=-\frac{(d-1)(d-3)}{4r^2}.$$

The residual vanishes for $d=1$ and $d=3$. Thus $e^{ipr/\hbar}/r$ is exact away from the source in three dimensions; in two dimensions $r^{-1/2}e^{ipr/\hbar}$ has a nonzero curvature residual and is only the large-$pr/\hbar$ limit of a Hankel wave.

2. A fold projection has $q-q_c=a\alpha^2$, with $a>0$, and nonzero smooth transported weight. Find the two branches and the isolated amplitude scaling as $q\downarrow q_c$.

Solution

The labels are $\alpha_\pm=\pm\sqrt{(q-q_c)/a}$ for $q>q_c$ and absent for $q<q_c$. Since $|dq/d\alpha|=2\sqrt{a(q-q_c)}$, each isolated amplitude scales as $(q-q_c)^{-1/4}$. An Airy approximation resolves their merger. The power comes from the projection Jacobian, even if the local momentum magnitude varies slowly.

References

• V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed. (Springer, 1989). Hamilton–Jacobi theory, characteristics, and symplectic geometry.
• M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics”, Reports on Progress in Physics 35, 315–397 (1972). Phase-integral methods, multidimensional rays, caustics, and uniform approximations.
• M. Born and E. Wolf, Principles of Optics, 7th expanded ed. (Cambridge University Press, 1999). Eikonal optics, ray transport, and optical caustics.
• M. Brack and R. K. Bhaduri, Semiclassical Physics, revised ed. (Westview Press, 2003). Practical multidimensional semiclassics, trace formulas, and examples.
• M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, 1990). Classical trajectories, semiclassical amplitudes, and nonintegrable dynamics.
• B. Helffer, Semi-Classical Analysis for the Schrödinger Operator and Applications, Lecture Notes in Mathematics 1336 (Springer, 1988). Multidimensional semiclassical constructions and tunneling from a mathematical perspective.
• R. G. Littlejohn, “The semiclassical evolution of wave packets”, Physics Reports 138, 193–291 (1986). Phase-space geometry, wave packets, metaplectic structure, and caustics.
• V. P. Maslov and M. V. Fedoriuk, Semi-Classical Approximation in Quantum Mechanics (D. Reidel, 1981). Canonical-operator treatment of Lagrangian manifolds and caustics.