Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Chapter 8. Section 8.4.
We’ve seen that the free-particle propagator can be obtained in the path integral approach by using only the classical path in the sum over paths. It turns out that it’s not too hard to calculate the propagator for a free particle properly, by summing over all possible paths. The notation used by Shankar is as follows.
We want to evaluate the path integral
The notation means an integration over all possible paths from to in the given time interval. This includes paths where the particle might move to the right for a while, then jog back to the left, then back to the right again and so on. This might seem like a hopeless task, but we can make sense of this method by splitting the time interval between and into small intervals, each of length . Thus an intermediate time , and the final time is .
For a free particle, there is no potential energy so the Lagrangian is just the kinetic energy:
We can estimate the velocity in each time slice by
Note that this assumes that the velocity within each time slice is constant, but as we make smaller and smaller, this is increasingly accurate. Also note that it is possible for to be both positive (if the particle moves to the right in the interval) or negative (if it moves to the left).
The action for a given path is given by the integral of the Lagrangian:
In our discretized approximation, we evaluate within each time slice, and becomes the interval length , so the action becomes a sum:
The key point here is to notice that we can label any given path by choosing values for all the s between the two times, and that each can vary independently of the others, over a range from to . We can therefore implement the multiple integration required by by integrating over all the variables separately. That is,
where is some constant to make the scale come out right.
We don’t integrate over or since these are fixed as the end points of the path. To get the final version, we need to take the limit of this expression as and . This still looks pretty scary, but in fact it is doable. We define the variable
This gives us
We can do the integral in stages in order to spot a pattern. Consider first the integral over , which involves only two of the factors in the integrand:
We first simplify the exponent
We can evaluate this using a standard Gaussian integral
To simplify the exponents on the RHS:
Thus we have
Having eliminated we can now do the integral over :
Again, we can simplify the exponent:
The integral now becomes
Doing the Gaussian integral on the RHS using 16:
Thus the combined integral over and is
The general pattern after integrations is (presumably this could be proved by induction, but we’ll accept the result):
where we reverted back to using 9.
Going back to 11, we must multiply the result by to get the final expression for the propagator:
In the limit as and , so we have
The expression we got earlier using the Schrödinger method is
Thus the full path integral gives the same result, with and (similarly for ), provided that we can set
Shankar then says that it is conventional to associate one factor of with each integration over an , and the remaining factor with the overall process. This seems to overlook a basic problem, in that as and , , so we seem to be cancelling two infinities when we multiply the path integral by .