To be used in future
 Section 5.2: Interpretation Up Chapter 5: Path Integrals Section 5.4: General Gaussian integrals 

5.3 Free particle revisited

It is quite instructive to evaluate the coordinate space path integral for a free particle directly, This is a Gaussian integral; we can in principle evaluate it by a change of variables, but there is an easier procedure: we can perform the integral, and then integrate over , We recognise a pattern. We can show that Thus, by induction, we find that in agreement with the result () above.
We know can interpret the exponent in a rather usueful way. The classical action is defined as the integral of the Lagrangian along the classical path, The path for the free particle is a straight line from to . The Lagrangian equals the kinetic energy, which is and thus we can reinterpret the result () as Results wehere we find that the effect of the sum over all paths can be summarised as giving rise to a prefactor time a phase factro containing the classicial action are very common in excat results for simple problems, and in approximatiuon schemes, as we shall see below.
 Section 5.2: Interpretation Up Chapter 5: Path Integrals Section 5.4: General Gaussian integrals