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