Semiclassical Propagator

The semiclassical propagator approximates quantum time evolution by a coherent sum over classical trajectories joining fixed endpoints. It is a position-space amplitude density, to be integrated against an initial wavefunction. Position eigenstates are not normalizable, and its squared modulus alone is not a transition probability.

Required background. Propagator kernels explain the integration measure and normalization; propagators and path integrals explain the fixed-endpoint action. The formula below uses flat Cartesian configuration coordinates, $T=t_b-t_a>0$, and a smooth scalar Hamiltonian $p^2/(2m)+V(q,t)$. Global topology and boundary conditions must be imposed separately, as in the circle example below.

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,

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

The Gaussian factor uses $i^{-d/2}=e^{-i\pi d/4}$, and $\nu_\gamma=0$ on a short nonfocal branch, increasing by conjugate-point multiplicity as time grows. This convention avoids counting the free-particle phase twice. 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 action variations are rapid on the $\hbar$ scale, nonstationary neighborhoods cancel asymptotically and stationary points of $S[q]$ supply the leading terms. An arbitrary large additive constant in the action does not control this approximation. 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,

$$q(t) = q_\gamma(t)+\eta(t), \qquad \eta(t_a)=\eta(t_b)=0,$$

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. With the initial position and both times fixed, define the stability block

$$B_\gamma=\left(\frac{\partial q_b}{\partial p_a}\right)_{q_a}, \qquad |D_\gamma|=|\det B_\gamma|^{-1}.$$

The inverse relation holds on a nonfocal branch. If $B_\gamma$ loses rank, nearby classical paths focus; the naive prefactor becomes singular and the semiclassical expression needs a uniform or distributional representation. The determinant and its caustic phases must be continued together, as explained in Maslov Index.

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. The action is quadratic in path fluctuations, so there are no higher fluctuation terms beyond the Gaussian integral.

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.

For example, a free particle on a circle of circumference $L$ with periodic boundary conditions, $\psi(x+L)=\psi(x)$, can wind any integer number of times. Choose representatives $x_a,x_b$ and lift each path to the line, where its final endpoint is $x_b+nL$ for $n\in\mathbb Z$. Each lift has its own action and contributes an amplitude. The winding sum imposes the global periodicity; a single Cartesian displacement $x_b-x_a$ would omit these branches. See Particle on a Ring for the spectral description.

The isolated-trajectory formula fails where the endpoint map loses rank and the Van Vleck prefactor diverges. A generic fold merges two branches; the oscillator focus is more degenerate and has an exact distributional kernel. Near such points, one replaces the isolated-trajectory sum by a uniform approximation, such as an Airy form for a simple fold caustic.

Validity and Failure Modes

The leading approximation requires isolated stationary paths and a nondegenerate quadratic fluctuation form, away from caustics. Relevant action variations must be large on the $\hbar$ scale; adding a large constant to the action does not improve the approximation.

Accuracy also depends on what is propagated and for how long. The initial packet and the measured observable must not resolve interference finer than the retained approximation controls. Long-time propagation in unstable systems can involve rapidly proliferating branches and growing errors, so a good short-time kernel does not establish accuracy at arbitrarily late times. Littlejohn’s wave-packet treatment discusses these limitations in phase space.

Turning points, separatrices, tunneling regions, singular potentials, and hard boundaries can require different local approximations or additional saddle contributions. In particular, real classical trajectories alone do not describe every tunneling contribution; complex or Euclidean saddles may be needed. These are limits of the isolated-path formula, not failures of the exact path integral.

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.

Exercises

1. A constant force has $V(x)=-Fx$. Find the endpoint action and Van Vleck determinant over a time $T>0$, and decide whether the semiclassical kernel is exact.

Solution

Using elapsed time $0\leq t\leq T$, the trajectory satisfying both endpoints is

$$x(t)=x_a+\left(\frac{x_b-x_a}{T}-\frac{FT}{2m}\right)t+\frac{F t^2}{2m}.$$

Integrating $L=m\dot x^2/2+Fx$ gives

$$S=\frac{m(x_b-x_a)^2}{2T} +\frac{FT}{2}(x_a+x_b)-\frac{F^2T^3}{24m}.$$

The mixed derivative gives $D=m/T$. Fluctuations have a purely quadratic action, so the Gaussian semiclassical kernel is exact. The force changes the phase and packet center, while the stability prefactor is the same as for free motion.

2. Verify the units of the $d$-dimensional kernel and explain why multiplying $|K|^2$ by a final volume does not produce a probability for an initial position eigenstate.

Solution

Each mixed action derivative has units action/length$^2$, so $(2\pi\hbar)^{-d/2}|D|^{1/2}$ has units length$^{-d}$. The evolved state is $\psi(q_b)=\int K(q_b,q_a)\psi(q_a)\,d^dq_a$. Position eigenstates are distributions and have no unit Hilbert-space norm; physical probabilities are obtained from a normalized initial packet and the final $|\psi|^2$.

3. Derive the endpoint momentum relations by varying the classical action while keeping the initial and final times fixed.

Solution

Integration by parts gives

$$\delta S_\gamma =\left[p\cdot\delta q\right]_{t_a}^{t_b} +\int_{t_a}^{t_b} \left(\frac{\partial L}{\partial q}-\frac{dp}{dt}\right) \cdot\delta q\,dt.$$

The integral vanishes on a classical path. Comparing the remaining boundary term $p_b\cdot\delta q_b-p_a\cdot\delta q_a$ with the endpoint differential of $S_\gamma$ gives $p_b=\partial S_\gamma/\partial q_b$ and $p_a=-\partial S_\gamma/\partial q_a$. Unlike the fluctuations used to evaluate a fixed-endpoint kernel, these variations deliberately move the endpoints between neighboring classical solutions.

4. For the free particle on a circle described above, find the action and Van Vleck determinant of each winding branch. Why does fixing endpoints not select one classical path?

Solution

On the covering line, the branch with winding number $n$ travels the displacement $x_b-x_a+nL$ in time $T$. Thus

$$S_n=\frac{m(x_b-x_a+nL)^2}{2T}, \qquad D_n=\frac{m}{T}.$$

Each integer gives a distinct stationary path satisfying the same physical endpoints on the circle. This is a boundary-value problem; the initial momentum is not fixed. The periodic kernel sums the corresponding complex free-particle amplitudes, with the usual distributional interpretation of a real-time kernel. Summing their squared magnitudes instead would discard interference and would not impose the correct periodic quantum evolution.

5. What does $\det B_\gamma=0$ signal, and why does the divergent isolated-path prefactor not imply an infinite probability for a wave packet?

Solution

The map from initial-momentum perturbations to final-position perturbations loses rank, so neighboring trajectories focus at the same endpoint. The inverse determinant in the Van Vleck prefactor becomes singular. A uniform approximation treats generic coalescing branches together; an exact focal kernel can instead have a distributional limit. For example, the harmonic oscillator at a half-period has a delta-function kernel implementing parity and a phase, as discussed in Van Vleck Determinant. Acting on a normalized packet remains well defined. The isolated-trajectory expression has ceased to be a valid ordinary function there, and its squared modulus cannot be interpreted as a probability density.

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.
• R. G. Littlejohn, “The semiclassical evolution of wave packets”, Physics Reports 138, 193–291 (1986).
• L. S. Schulman, Techniques and Applications of Path Integration, Wiley, 1981.