Van Vleck Determinant

The Van Vleck determinant is the classical stability determinant that supplies the leading prefactor in the semiclassical propagator. It measures how a narrow bundle of nearby classical trajectories launched from one endpoint spreads or focuses at the other endpoint. Multidimensional WKB Preview gives the corresponding local ray-tube transport picture; for the broader stationary-phase and action-scaling mechanism, see Semiclassical Limit.

For a classical trajectory $\gamma$ from $(q_a,t_a)$ to $(q_b,t_b)$, let

$$S_\gamma(q_b,t_b;q_a,t_a) = \int_{t_a}^{t_b} L(q_\gamma,\dot q_\gamma,t)\,dt$$

be Hamilton’s principal function. The Van Vleck determinant is

$$D_\gamma = \det\left( - \frac{\partial^2S_\gamma} {\partial q_b\,\partial q_a} \right).$$

The corresponding semiclassical propagator has prefactor

$$\left( \frac{1}{2\pi i\hbar} \right)^{d/2} \lvert D_\gamma\rvert^{1/2}.$$

The determinant is the multidimensional generalization of the $1/\sqrt{p(x)}$ WKB amplitude factor: both express conservation of probability flux along families of classical trajectories.

Endpoint Derivatives

Hamilton’s principal function generates the endpoint momenta:

$$p_b = \frac{\partial S_\gamma}{\partial q_b}, \qquad p_a = - \frac{\partial S_\gamma}{\partial q_a}.$$

Differentiating the second relation with respect to $q_b$ gives

$$\frac{\partial p_a}{\partial q_b} = - \frac{\partial^2S_\gamma} {\partial q_b\,\partial q_a}.$$

Thus

$$D_\gamma = \det\left( \frac{\partial p_a}{\partial q_b} \right),$$

with $q_a$, $t_a$, and $t_b$ held fixed. This form makes clear that the determinant relates final position resolution to initial momentum resolution.

Trajectory Density

Fix $q_a$ and vary the initial momentum $p_a$. The classical equations of motion define a map

$$p_a \longmapsto q_b(q_a,p_a;T), \qquad T=t_b-t_a.$$

Let

$$B = \frac{\partial q_b}{\partial p_a}$$

be the block of the classical stability matrix that maps initial momentum perturbations into final position perturbations. Where this block is invertible,

$$\lvert D_\gamma\rvert = \lvert \det B \rvert^{-1}.$$

If a small range of initial momenta maps to a large range of final positions, the trajectory density at $q_b$ is low and the prefactor is small. If trajectories focus, $\det B$ becomes small and the prefactor grows.

Caustics

A caustic occurs when the map from initial momentum to final position becomes singular:

$$\det B=0.$$

Equivalently, the Van Vleck determinant diverges in the isolated-trajectory formula. The exact quantum propagator does not usually diverge there. The divergence means that the chosen semiclassical chart has failed, and multiple classical branches must be combined by a uniform approximation.

Crossing a caustic also changes the phase of the square root in the prefactor. The Maslov Index records this phase change.

Free-Particle Example

For a free particle in $d$ dimensions over time $T>0$,

$$S_{\mathrm{free}} = \frac{m\lvert q_b-q_a\rvert^2}{2T}.$$

Therefore

$$- \frac{\partial^2S_{\mathrm{free}}} {\partial q_b\,\partial q_a} = \frac{m}{T}I_d,$$

and

$$D_{\mathrm{free}} = \left( \frac{m}{T} \right)^d.$$

The prefactor becomes

$$\left( \frac{m}{2\pi i\hbar T} \right)^{d/2},$$

which is the exact free-particle normalization.

Harmonic-Oscillator Example

For a one-dimensional harmonic oscillator over time $T$ with $\sin\omega T\ne0$, the classical action is

$$S_{\mathrm{osc}} = \frac{m\omega}{2\sin\omega T} \left[ (q_b^2+q_a^2)\cos\omega T - 2q_aq_b \right].$$

The mixed second derivative gives

$$- \frac{\partial^2S_{\mathrm{osc}}} {\partial q_b\,\partial q_a} = \frac{m\omega}{\sin\omega T}.$$

Thus

$$\lvert D_{\mathrm{osc}}\rvert^{1/2} = \left\lvert \frac{m\omega}{\sin\omega T} \right\rvert^{1/2}.$$

The determinant diverges when

$$\sin\omega T=0,$$

which is exactly when oscillator trajectories refocus. These are caustic times, and the Maslov phase must be updated as they are crossed.

Relation to Stability Matrices

The linearized classical flow can be written as

$$\begin{pmatrix} \delta q_b\\ \delta p_b \end{pmatrix} = \begin{pmatrix} A & B\\ C & D \end{pmatrix} \begin{pmatrix} \delta q_a\\ \delta p_a \end{pmatrix}.$$

The Van Vleck determinant is controlled by the $B$ block when $q_a$ and $T$ are fixed:

$$\lvert D_\gamma\rvert = \lvert\det B\rvert^{-1}.$$

This is why the determinant is often called a stability determinant. It knows whether neighboring trajectories separate, focus, or form caustics.

Common Mistakes

• Treating the Van Vleck determinant as a quantum correction independent of classical mechanics.
• Dropping the absolute value without tracking the associated Maslov phase.
• Using the isolated-trajectory formula at caustics where $\det B=0$.
• Confusing the determinant with the full monodromy matrix rather than a particular endpoint block.
• Forgetting that dimensions matter: the determinant carries units needed to normalize the propagator.

Exercises

1. Show that the free-particle determinant equals $(m/T)^d$.

Solution

The free-particle action is

$$S_{\mathrm{free}} = \frac{m}{2T} (q_b-q_a)\cdot(q_b-q_a).$$

Taking one derivative with respect to $q_b$ and one with respect to $q_a$ gives

$$\frac{\partial^2S_{\mathrm{free}}} {\partial q_{b,i}\,\partial q_{a,j}} = - \frac{m}{T}\delta_{ij}.$$

Therefore

$$D_{\mathrm{free}} = \det\left( \frac{m}{T}I_d \right) = \left( \frac{m}{T} \right)^d.$$

2. For one-dimensional motion, explain why a zero of $\partial q_b/\partial p_a$ is a caustic.

Solution

If $\partial q_b/\partial p_a=0$, then nearby initial momenta land at the same final position to first order. The trajectory map from initial momentum to final position is singular, so the density of classical paths projected into configuration space diverges. The isolated semiclassical prefactor, proportional to $\lvert\partial q_b/\partial p_a\rvert^{-1/2}$, also diverges. This is the signature of a caustic.

3. Locate the caustic times of the one-dimensional harmonic oscillator from the Van Vleck determinant.

Solution

For the oscillator,

$$D_{\mathrm{osc}} = \frac{m\omega}{\sin\omega T}.$$

The isolated-trajectory prefactor is singular when $\sin\omega T=0$. Therefore the caustic times are

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

References

• J. H. Van Vleck, “The correspondence principle in the statistical interpretation of quantum mechanics,” Proceedings of the National Academy of Sciences 14, 178-188, 1928.
• M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer, 1990.
• 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.