Bring the chapter together: the propagator and its composition, time slicing, the sum over histories, the classical limit by stationary phase, the free particle and the oscillator done exactly, and imaginary time. No worked example sits above the answers.

A river delta splits one river into countless channels that meet again at the sea; the water that arrives is the sum of all of them. The propagator is that sum, taken over the histories of a particle.

$K=\int\mathcal{D}x\,e^{iS/\hbar}$ with $S=\int(\tfrac12m\dot{x}^{2}-V)\,dt$; $K_{\text{free}}=\sqrt{\frac{m}{2\pi i\hbar T}}\,e^{im(x_{b}-x_{a})^{2}/2\hbar T}$.

• $\delta S=0$ gives Newton’s law; as $\hbar\to0$, $K\approx A\,e^{iS_{\text{cl}}/\hbar}$.
• For quadratic actions the path integral is exact: the classical action times a factor of $T$.
• In imaginary time the weights are $e^{-S_{E}/\hbar}$, and the long-time decay gives $E_{0}$.