Maslov Index

The Maslov index is an integer phase correction in semiclassical mechanics. It records how many caustics, turning points, or conjugate points a semiclassical branch has crossed, with a convention-dependent sign. In the convention used here, each unit of Maslov index contributes the phase factor

$$\exp\left( - i\frac{\pi}{2} \right)$$

to a transported semiclassical branch.

The index is the bookkeeping device behind the $1/2$ in one-dimensional WKB quantization and the phase factor in semiclassical propagators. It is not an extra empirical rule. It is what remains when a local WKB branch is continued through places where its naive amplitude would become singular.

Why a Correction Is Needed

The leading WKB form in an allowed region is

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

At a turning point $p(x)=0$, this expression fails. The exact local equation is repaired by Airy matching, as described on Turning Points and Connection Formulas. The result is a phase shift of $\pi/4$ for the standing wave attached to a decaying forbidden tail.

When the wave is carried around a complete closed orbit, these local phase losses accumulate. The Maslov index packages the accumulated caustic phase into a single integer.

One-Dimensional Smooth Well

For a smooth one-dimensional bound state, the classical closed orbit has two turning points. The EBK phase condition can be written

$$\frac{1}{\hbar} \oint p\,dx - \frac{\pi}{2}\mu = 2\pi n.$$

The two ordinary turning points give

$$\mu=2.$$

Therefore

$$\oint p\,dx = 2\pi\hbar \left( n+\frac12 \right).$$

This is the modern WKB version of the Bohr-Sommerfeld rule. The $1/2$ is the two-turning-point Maslov correction.

Caustics and Branch Changes

A caustic is a place where the projection from a Lagrangian manifold in phase space down to configuration space becomes singular. In one-dimensional WKB this is visible as a turning point. In multidimensional semiclassics, caustics occur when nearby classical trajectories focus, so the configuration-space amplitude predicted by a single branch diverges.

The divergence is not physical. It signals that the chosen coordinate representation is a bad local description of the semiclassical wave. Passing through the caustic changes the branch of the square root in the semiclassical amplitude, and that branch change is tracked by the Maslov phase.

This is closely analogous to geometrical optics. Ray amplitudes diverge at focal caustics, but the wave field remains finite when described by an appropriate uniform approximation.

EBK Convention

For an integrable system, the EBK rule around a cycle $\gamma_j$ is

$$J_j = \oint_{\gamma_j}p\cdot dq = 2\pi\hbar \left( n_j+\frac{\mu_j}{4} \right).$$

Equivalently,

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

This equation is often the least ambiguous way to remember the convention. If another source defines the Maslov index with the opposite sign, the final quantization rule can still agree after translating conventions.

Propagator Convention

For the semiclassical propagator, the Maslov index appears as a phase attached to each classical trajectory:

$$K_{\mathrm{sc}}(q_b,t_b;q_a,t_a) \sim \sum_\gamma A_\gamma \exp\left[ \frac{i}{\hbar}S_\gamma - i\frac{\pi}{2}\nu_\gamma \right].$$

Here $\nu_\gamma$ counts conjugate points along the trajectory in the propagator convention. The same geometric phenomenon is being measured, but notation varies across EBK, WKB, Morse theory, and trace-formula literature. The safe habit is to state the phase convention along with the index.

Harmonic-Oscillator Example

The one-dimensional harmonic oscillator propagator has caustics at times

$$T = \frac{k\pi}{\omega}, \qquad k\in\mathbb Z,$$

because all trajectories refocus after half periods. Between caustics the semiclassical expression is smooth, but crossing a caustic requires a Maslov phase jump.

For bound-state quantization, the same oscillator has two turning points on each full orbit, so $\mu=2$. The WKB rule gives

$$E_n = \hbar\omega \left( n+\frac12 \right),$$

which is exact for the oscillator.

Common Mistakes

• Treating the Maslov index as a decorative correction rather than a phase needed for single-valuedness.
• Assuming every closed cycle has $\mu=2$.
• Mixing EBK and propagator conventions without checking signs.
• Thinking the WKB divergence at a caustic is a physical divergence.
• Ignoring caustic phases when comparing semiclassical formulas across different coordinate representations.

Cross-Links

• Action and Phase
• Turning Points and Connection Formulas
• EBK Quantization
• Semiclassical Propagator
• Van Vleck Determinant

Exercises

1. Use the EBK phase condition to show that $\mu=2$ gives the usual one-dimensional WKB shift.

Solution

The phase condition is

$$\frac{J}{\hbar} - \frac{\pi}{2}\mu = 2\pi n.$$

With $\mu=2$,

$$\frac{J}{\hbar} - \pi = 2\pi n,$$

so

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

2. A cycle is a smooth rotation angle with no caustic. What Maslov shift should be used in EBK?

Solution

If the cycle has no caustic and no turning point, the Maslov index for that cycle is $\mu=0$ in the convention used here. The EBK condition is then $J=2\pi\hbar n$ with no $1/2$ shift.

3. Why does a caustic require a special approximation even when the exact wavefunction is finite?

Solution

A caustic is a singularity of the chosen semiclassical representation, not necessarily of the exact wavefunction. The local WKB amplitude contains a Jacobian or momentum factor that becomes singular. A uniform approximation, such as Airy matching at a simple turning point, replaces the singular local expression by a finite wave solution and determines the correct phase connection.

References

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