Semiclassical Propagator

The semiclassical propagator is the leading stationary-phase approximation to the quantum time-evolution kernel. It expresses the amplitude to travel from $q_a$ at time $t_a$ to $q_b$ at time $t_b$ as a sum over classical trajectories with those endpoints. The coordinate-momentum background is Phase Space. For the Core preview, see Semiclassical Limit Overview; for the Toolkit-level stationary-phase mechanism, see Semiclassical Limit.

The applied roadmap connecting this trajectory formula to local WKB, turning points, tunneling, and quantization is WKB and Semiclassical Methods.

The exact propagator is

$$K(q_b,t_b;q_a,t_a) = \langle q_b\vert U(t_b,t_a)\vert q_a\rangle.$$

In the semiclassical approximation,

$$K_{\mathrm{sc}}(q_b,t_b;q_a,t_a) = \sum_{\gamma:a\to b} \left( \frac{1}{2\pi i\hbar} \right)^{d/2} \lvert D_\gamma \rvert^{1/2} \exp\left[ \frac{i}{\hbar}S_\gamma - i\frac{\pi}{2}\nu_\gamma \right].$$

Here $d$ is the number of configuration-space dimensions, $S_\gamma$ is the classical action along trajectory $\gamma$, $D_\gamma$ is the Van Vleck determinant, and $\nu_\gamma$ is a Maslov index. The Van Vleck Determinant page explains the prefactor in detail.

Path Integral Origin

The path-integral representation of the propagator has the schematic form

$$K(q_b,t_b;q_a,t_a) = \int_{q(t_a)=q_a}^{q(t_b)=q_b} \mathcal Dq\, \exp\left[ \frac{i}{\hbar}S[q] \right].$$

When the action is large compared with $\hbar$, the integral is dominated by stationary points of $S[q]$. These are the classical paths satisfying the Euler–Lagrange equations with fixed endpoints. The mechanics background is reviewed in Lagrangian Mechanics Review, and the finite-dimensional approximation logic is explained in Asymptotic Analysis:

$$\delta S[q_\gamma]=0.$$

Expanding around a classical path,

$$q(t) = q_\gamma(t)+\eta(t),$$

gives

$$S[q] = S[q_\gamma] + \frac12 \delta^2S[\eta,\eta] + \cdots.$$

The linear term vanishes by the classical equations of motion. The Gaussian integral over fluctuations gives the prefactor. The phase of that Gaussian, including changes at caustics, gives the Maslov factor.

Classical Action

For a trajectory $\gamma$ with endpoints fixed, the classical action is Hamilton’s principal function:

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

It generates the endpoint momenta:

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

These relations connect the propagator phase to the Hamilton–Jacobi description of classical mechanics. The Hamilton–Jacobi Theory Preview gives the wavefunction-phase derivation and stationary composition law. The endpoint relations also explain why the second derivatives of $S_\gamma$ determine the amplitude.

Prefactor and Stability

The Van Vleck determinant in the convention used here is

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

It measures how nearby initial momenta spread into final positions. If nearby classical paths focus, the determinant becomes large; if a caustic is reached, the naive prefactor becomes singular and the semiclassical expression needs a uniform repair.

This is the propagator analogue of the $1/\sqrt{p(x)}$ amplitude in one-dimensional WKB. Both are conservation-of-flux factors for families of classical paths.

Free-Particle Check

For a free particle in $d$ dimensions over time $T=t_b-t_a\gt0$,

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

The mixed second derivative is

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

so

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

The semiclassical propagator becomes

$$K_{\mathrm{free}}(q_b,T;q_a,0) = \left( \frac{m}{2\pi i\hbar T} \right)^{d/2} \exp\left[ \frac{i}{\hbar} \frac{m\lvert q_b-q_a\rvert^2}{2T} \right],$$

which is the exact free-particle kernel.

Multiple Classical Paths

For some endpoints and times, more than one classical path connects $q_a$ to $q_b$. The semiclassical propagator is then a coherent sum over branches. Interference between branches is a physical prediction, not a nuisance.

The approximation is least reliable near caustics, where two or more branches coalesce and the Van Vleck prefactor diverges. Near such points, one replaces the isolated-trajectory sum by a uniform approximation, such as an Airy form for a simple fold caustic.

QFT Bridge

The semiclassical propagator is the finite-dimensional quantum-mechanical prototype of saddle-point expansions in path integrals. In field theory, the integration variable is a field configuration rather than a particle path, and the saddle points are classical field configurations. The same pattern appears:

$$\text{phase} \sim \frac{S_{\mathrm{classical}}}{\hbar}, \qquad \text{prefactor} \sim \text{fluctuation determinant}.$$

The quantum-mechanical case is the cleanest place to learn what is meant by stationary phase, fluctuation determinants, and caustic phases before those ideas are used in field-theoretic settings.

Common Mistakes

• Reading the sum over classical paths as a classical probability sum. It is an amplitude sum with phases.
• Dropping the prefactor when comparing with exact kernels or normalization.
• Forgetting the Maslov phase at caustics.
• Assuming the stationary-phase approximation is uniform near branch coalescence.
• Treating the path-integral expression as a literal ordinary integral without specifying its limiting or discretized meaning.

Exercises

1. Derive the free-particle Van Vleck determinant in $d$ dimensions.

Solution

For

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

one has

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

Therefore

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

and

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

2. Explain why more than one classical trajectory can contribute to the same propagator.

Solution

The boundary-value problem fixes the initial point, final point, and travel time, but it does not always have a unique classical solution. In a potential with turning motion, on a curved configuration space, or after sufficiently long times, several classical paths may connect the same endpoints. Stationary phase then gives one contribution from each stationary path, and the quantum result is their coherent sum.

3. Why does the semiclassical propagator need a Maslov phase?

Solution

The Gaussian fluctuation determinant changes phase when a trajectory crosses a caustic or conjugate point. If that phase is not tracked, the semiclassical kernel has the wrong branch and fails to match across focal points. The Maslov index records the required phase changes in integer units of $\pi/2$.

References

• 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.
• L. S. Schulman, Techniques and Applications of Path Integration, Wiley, 1981.