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 $I_j=J_j/(2\pi)$,

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

Required background. Turning-point matching explains the local phase shifts. Here the scalar Hamiltonian has no additional order-$\hbar$ subprincipal phase or internal Berry holonomy; such phases must be included when present.

Why Tori Appear

A system with $d$ degrees of freedom is integrable when it has $d$ independent constants of motion in involution. A connected compact regular common level set of these integrals is a $d$-torus under the Liouville–Arnold hypotheses. Boundedness alone does not exclude a singular level set such as a separatrix. 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,
• perturbations of resonant tori, where small denominators can obstruct an expansion around the integrable system,
• singular endpoints, such as radial centrifugal barriers,
• nonintegrable regions where invariant tori are destroyed,
• higher or coalescing caustics requiring more than ordinary Maslov matching.

A rational frequency ratio is not itself a failure of EBK: the separable oscillator above remains valid even when $\omega_x=\omega_y$. Ordinary caustics are already accommodated by the Maslov indices.

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.

Exercises

1. For the isotropic two-dimensional oscillator, use EBK to find the energy and degeneracy at total excitation $N=n_x+n_y$. Does the rational frequency ratio invalidate the result?

Solution

Both actions have $\mu=2$, giving $E_N=\hbar\omega(N+1)$. There are $N+1$ nonnegative pairs $(n_x,n_y)$ with sum $N$. This exact resonant example shows that rational frequency ratios do not by themselves spoil quantization. A perturbation coupling the degenerate states requires a separate analysis.

2. A scalar wave on a ring obeys $\psi(\phi+2\pi)=e^{i\beta}\psi(\phi)$. With no caustic, find its allowed angular momenta.

Solution

The phase condition is $2\pi p_\phi/\hbar=2\pi n+\beta$, so $p_\phi=\hbar(n+\beta/(2\pi))$. The shift comes from the boundary holonomy, not from a Maslov index. In electromagnetic applications distinguish canonical momentum from mechanical momentum before interpreting it as a rotation speed.

References

• M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics”, Reports on Progress in Physics 35, 315-397, 1972.
• M. Brack and R. K. Bhaduri, Semiclassical Physics, Westview Press, 2003.
• M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer, 1990.
• J. B. Keller, “Corrected Bohr–Sommerfeld Quantum Conditions for Nonseparable Systems”, Annals of Physics 4, 180–188 (1958).
• V. P. Maslov and M. V. Fedoriuk, Semi-Classical Approximation in Quantum Mechanics, Reidel, 1981.