Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 304 / 1 / a / ii / Solution

The Euler-Lagrange equation is $m\ddot{x}=0$. The unique path with the prescribed endpoints is thereforeIt is a minimum of the Euclidean action and a stationary point of the real-time action. Its classical action isThe principle of stationary action consequently fixes the position-dependent phase of the semiclassical propagator asBecause the action is quadratic, the stationary-phase evaluation of the path integral is exact. Composition of propagators, or the Van Vleck determinant, gives $C(T)=\sqrt{m/(2\pi i\hslash T)}$, reproducing part i.