# The Free-Particle Propagator

### From FSUPhysicsWiki

We will now evaluate the kernel for a free particle. In this case, the action is just

Note that we renamed to and to the reason for this will become clear shortly. Let us now discretize the path that the particle takes, so that the intermediate positions are We discretize the time axis similarly, with a spacing between two subsequent times, so that and so on. The action may then be written as

The kernel now becomes

We will now evaluate this integral. Let us first switch to the variables,

We then get

Although the multiple integral looks formidable, it is not. Let us begin by doing the integral over Considering just the part of the integrand that involves we get

Now let us evaluate the integral over Again considering just the part of the integrand that involves we get

We now continue to do this until all of the have been integrated out. At the step (i.e., integrating out ), the integral that we evaluate and the solution are

Combining all of these results together, we find that the kernel is

or, rewriting in terms of and

Since we divided the time interval up into equal amounts, we note that We may now take the limit, finally obtaining the free-particle propagator,