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.

Required background. The semiclassical propagator fixes the Cartesian kernel measure and phase convention.

Endpoint derivatives and matrix orientation

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

$$\frac{\partial p_{a,j}}{\partial q_{b,i}} =-\frac{\partial^2S_\gamma}{\partial q_{b,i}\partial q_{a,j}}.$$

With row indices labeling output components, these two matrices are transposes; their determinants agree. 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 divergence signals failure of this isolated-trajectory chart. A generic fold needs a uniform approximation. At an exact oscillator focus the kernel itself is distributional: $K(x_b,\pi/\omega;x_a,0)=-i\delta(x_b+x_a)$. Regular evolved wavefunctions remain well defined after the kernel is integrated against the initial state.

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

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

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

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

The full classical flow matrix is symplectic and has determinant one, even when its $B$ block is singular. The symbol $D_{\mathrm{cl}}$ for its lower-right block is unrelated to the scalar Van Vleck determinant $D_\gamma$.

Exercises

1. For the inverted oscillator $V=-m\Omega^2x^2/2$, compute $B=\partial x_b/\partial p_a$ at fixed $x_a$. Does exponential instability produce a real positive-time caustic?

Solution

Classical motion gives

$$x_b=x_a\cosh\Omega T+\frac{p_a}{m\Omega}\sinh\Omega T, \qquad B=\frac{\sinh\Omega T}{m\Omega}.$$

Thus $D=m\Omega/\sinh\Omega T$ and the prefactor decays as $e^{-\Omega T/2}$ at long positive times. There is no positive-time zero of $B$. Instability causes spreading here; it is not synonymous with focusing.

2. Check the free-particle composition law for the determinant in one dimension, using a stationary integral over an intermediate position at times $T_1,T_2>0$.

Solution

The combined phase has second derivative $m/T_1+m/T_2$ with respect to the intermediate position. Its Gaussian integration factor has magnitude $[2\pi\hbar/(m/T_1+m/T_2)]^{1/2}$. Multiplying the two kernel prefactors yields $[m/(2\pi\hbar(T_1+T_2))]^{1/2}$, exactly the prefactor for the total time. The Gaussian phase likewise supplies the correct $e^{-i\pi/4}$ rather than two independent copies of it.

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. H. Van Vleck, “The correspondence principle in the statistical interpretation of quantum mechanics”, Proceedings of the National Academy of Sciences 14, 178-188, 1928.