Understanding Non-Relativistic Path Integral Formulation

[CPL.Luke]

I'm doing a project for my quantu class on the non-relativistic path integral formulation. I took out "quantum mechanics and path integrals" feynmann, but he doesn't seem to like explaining explicitly how certain results are obtained...

so my two main questions are should the weight function exp[i/h S] be computed between the time of first measurement and the time of second measurement?

and also how do you put the integral into terms of x1,x2,x3... I don't really understand how these are defined, in the book he seemed to be describing them as points for a riemann sum but then he wrote an integral with respect to all of them and I got lost.


do you merely commput the action for some path x1(t) integrate and then attempt to integrate then multiply it by the weight function comuted for x2 and integrate with respect to x2 and simply continue this process until your done?

 

[unica]

You can see this book:Quantum Field Theory,Lewis H.Ryder.It may help you to understand your questions

 

[Meir Achuz]

There is an old QM text by Beard that gives a silmple treatment of the path integral approach.

 

[CPL.Luke]

I'm primarily looking for a book that has at least one example of some simple case worked out in gory detail, just because rightnow while I know what the equation for a path integral is I have absolutely no idea how to compute it for anything.

do either of those books have the soething like that in them.

 

[OOO]

so my two main questions are should the weight function exp[i/h S] be computed between the time of first measurement and the time of second measurement?

Yes. For position measurements. What you actually want to do is:
1) start with a localized particle
2) let it evolve for a small time
3) examine where the particle could be now
4) for every place where it could be, repeat from step 2
5) ...until you reach your destination time
6) where you judge the probability amplitude to find the particle localized again somewhere

and also how do you put the integral into terms of x1,x2,x3... I don't really understand how these are defined, in the book he seemed to be describing them as points for a riemann sum but then he wrote an integral with respect to all of them and I got lost.

By means of the completeness relation for position eigenstates. This basically says: if the particle goes from x1 to x2 to x3 then the probability amplitude to go from x1 to x3 without knowing where x2 is, is a sum over all x2. It's simple: since you don't know where x2 is, you have to take every possibility into account.

do you merely commput the action for some path x1(t) integrate and then attempt to integrate then multiply it by the weight function comuted for x2 and integrate with respect to x2 and simply continue this process until your done?

You can think of it this way, but I guess this would be mathematically ill-defined. The (relatively) safe way to do the path integral is by the time slices (=lattice approximation) Feynman or every other textbook explains. Somehow one's considering all paths at once.

 

[CPL.Luke]

ah that helps a lot, I'll go back and tr to work out the free particle again, and see if I can get the books answer