EBK Quantization

EBK quantization is the semiclassical quantization rule for classically integrable systems. It generalizes the one-dimensional Bohr-Sommerfeld Quantization condition from a single closed orbit to motion on an invariant torus. Multidimensional WKB Preview develops the local eikonal branches and transport amplitudes that must be assembled before this global phase-consistency rule can be imposed.

The name abbreviates Einstein-Brillouin-Keller. The modern rule can be stated in terms of action variables. For each independent cycle $\gamma_j$ on a $d$-dimensional invariant torus,

$$J_j = \oint_{\gamma_j} p\cdot dq, \qquad j=1,\ldots,d.$$

EBK quantization says

$$J_j = 2\pi\hbar \left( n_j+\frac{\mu_j}{4} \right), \qquad n_j\in\mathbb Z,$$

where $\mu_j$ is the Maslov index of the cycle. Equivalently, with the action variable $I_j=J_j/(2\pi)$,

$$I_j = \hbar \left( n_j+\frac{\mu_j}{4} \right).$$

This page is the canonical home for the EBK rule in this volume. The Maslov Index page explains the phase correction, and the Semiclassical Propagator page explains the related stationary-phase approximation to time evolution.

Why Tori Appear

A system with $d$ degrees of freedom is integrable when it has $d$ independent constants of motion in involution. Under regularity assumptions, bounded motion lies on a $d$-torus in phase space. On that torus the motion is described by action-angle coordinates:

$$(q,p) \longrightarrow (I_1,\ldots,I_d;\theta_1,\ldots,\theta_d),$$

with Hamiltonian

$$H=H(I_1,\ldots,I_d).$$

The angles advance linearly in time,

$$\dot{\theta}_j = \omega_j(I) = \frac{\partial H}{\partial I_j}.$$

The actions are constants of the classical motion. Semiclassically, the quantum wavefunction must be single-valued after it is transported around every independent cycle of the torus. This single-valuedness is what quantizes the actions.

Phase Around a Cycle

Along a semiclassical branch, the phase is the classical action divided by $\hbar$:

$$\psi \sim A\,e^{iS/\hbar}.$$

When the branch is continued around a cycle $\gamma_j$, the action phase changes by

$$\frac{1}{\hbar} \oint_{\gamma_j} p\cdot dq = \frac{J_j}{\hbar}.$$

If that were the whole story, single-valuedness would require $J_j=2\pi\hbar n_j$. But semiclassical wavefunctions also acquire phase shifts at caustics and turning points. With the Maslov correction included, the phase consistency condition is

$$\frac{J_j}{\hbar} - \frac{\pi}{2}\mu_j = 2\pi n_j.$$

Solving gives the EBK rule

$$J_j = 2\pi\hbar \left( n_j+\frac{\mu_j}{4} \right).$$

Different texts sometimes place signs and orientations differently in the definition of $\mu_j$. The quantization rule above fixes the convention used here.

One-Dimensional Check

For a smooth one-dimensional bound state with two turning points, there is one closed phase-space cycle. The action is

$$J = \oint p\,dx = 2 \int_{x_1}^{x_2} \sqrt{2m(E-V(x))} \,dx.$$

The closed cycle crosses two ordinary caustics, so $\mu=2$. EBK gives

$$J = 2\pi\hbar \left( n+\frac12 \right),$$

or

$$\int_{x_1}^{x_2} \sqrt{2m(E-V(x))} \,dx = \pi\hbar \left( n+\frac12 \right).$$

This is exactly the WKB bound-state rule derived on the Bohr-Sommerfeld page. EBK does not replace the one-dimensional rule; it explains it as the one-cycle case.

Separable Oscillator Example

For a two-dimensional separable harmonic oscillator,

$$H = \frac{p_x^2}{2m} + \frac12m\omega_x^2x^2 + \frac{p_y^2}{2m} + \frac12m\omega_y^2y^2,$$

the invariant torus has two independent cycles, one for each oscillator. The classical actions are

$$J_x=\frac{2\pi E_x}{\omega_x}, \qquad J_y=\frac{2\pi E_y}{\omega_y}.$$

Each oscillator cycle has two turning points, so

$$\mu_x=\mu_y=2.$$

The EBK conditions are

$$\frac{2\pi E_x}{\omega_x} = 2\pi\hbar \left( n_x+\frac12 \right), \qquad \frac{2\pi E_y}{\omega_y} = 2\pi\hbar \left( n_y+\frac12 \right).$$

Thus

$$E = \hbar\omega_x \left( n_x+\frac12 \right) + \hbar\omega_y \left( n_y+\frac12 \right).$$

As in one dimension, the harmonic oscillator is special: the leading semiclassical result gives the exact spectrum.

Cyclic Coordinates

Not every action has a turning-point shift. If $\phi$ is a cyclic angle with conjugate momentum $p_\phi$, then a closed angular cycle has action

$$J_\phi = \oint p_\phi\,d\phi = 2\pi p_\phi.$$

For a smooth full rotation with no caustic on the cycle,

$$\mu_\phi=0,$$

so EBK gives

$$p_\phi = n_\phi\hbar.$$

This illustrates why the Maslov term is not a universal $1/2$ shift. The shift depends on the geometry of the cycle.

Limitations

EBK is an integrable-system rule. It assumes that the relevant classical motion is organized by invariant tori and that the action variables can be defined globally or at least patchwise with controlled singularities.

The rule becomes delicate near:

• separatrices, where classical periods can diverge,
• resonances, where frequency ratios are rational,
• singular endpoints, such as radial centrifugal barriers,
• nonintegrable regions where invariant tori are destroyed,
• caustics where a single WKB branch is not enough.

For chaotic systems there is no global set of EBK actions. Semiclassical quantization then uses different tools, such as trace formulas built from periodic orbits. That is a different semiclassical regime, not a small correction to EBK.

For correspondence with a fixed classical torus, the quantum numbers and $\hbar$ must be scaled jointly so that the action variables remain fixed. Semiclassical Limits and Correspondence derives how this limit turns neighboring EBK energy differences into harmonics of the classical frequencies.

Common Mistakes

• Applying EBK to a nonintegrable system without checking whether invariant tori exist.
• Treating every cycle as if it had the one-dimensional $1/2$ shift.
• Confusing $J_j=\oint p\cdot dq$ with $I_j=J_j/(2\pi)$.
• Forgetting that action variables depend on the chosen independent cycles.
• Using EBK near a separatrix without a uniform approximation.

Exercises

1. Show that the EBK rule reduces to the WKB bound-state rule for a one-dimensional smooth well.

Solution

For a one-dimensional well,

$$J = \oint p\,dx = 2 \int_{x_1}^{x_2}p(x)\,dx.$$

Two smooth turning points give $\mu=2$. EBK gives

$$J = 2\pi\hbar \left( n+\frac{\mu}{4} \right) = 2\pi\hbar \left( n+\frac12 \right).$$

Dividing by $2$ gives

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

which is the WKB quantization rule.

2. For a cyclic coordinate $\phi$ with constant $p_\phi$, evaluate $J_\phi$ and state the EBK condition when $\mu_\phi=0$.

Solution

The action around the full angular cycle is

$$J_\phi = \int_0^{2\pi}p_\phi\,d\phi = 2\pi p_\phi.$$

With $\mu_\phi=0$,

$$2\pi p_\phi = 2\pi\hbar n_\phi,$$

so $p_\phi=n_\phi\hbar$.

3. Explain why EBK cannot be used as a global quantization rule for a generic chaotic bound system.

Solution

EBK requires $d$ independent actions associated with cycles on invariant tori. A generic chaotic system does not have global invariant tori filling the relevant energy surface. Its trajectories are not organized by action-angle variables, so the EBK input data are missing. Semiclassical approaches to chaotic systems instead use periodic-orbit information and trace formulas.

References

• M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer, 1990.
• V. P. Maslov and M. V. Fedoriuk, Semi-Classical Approximation in Quantum Mechanics, Reidel, 1981.
• M. Brack and R. K. Bhaduri, Semiclassical Physics, Westview Press, 2003.
• M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Reports on Progress in Physics 35, 315-397, 1972.