How Does Schwinger's Method Reveal Information About Feynman's Propagator?

[Morgoth]

Well I am doing an assignment concerning methods of finding Feynman's propagator.

I understand pretty well how everything's coming up. I also get that the form:
<x'',t'|x',0> defined as Feynman's propagator gives the amplitude of a system initially being in state |x',0> to be in |x'',t> after time t.

What I don't get is after you use Schwinger's Method to calculate Feynman's propagator for the Harmonic Oscilator you end up in a form:
<x'',t'|x',0>=SQRT[mω/2πihsin(ωt)] * exp{ (imω/2hsin(ωt))* [(x''^2- x'^2)cos(ωt)-2x''x'] }

(Sorry for the not beautiful formula).

In fact I cannot understand what kind of information the above form gives us.
+
For the amplitude the change of x''->x' and x'->x'' would be a symmetry since they are both into the exp[if] which is just a phase factor?

Can anyone give me an idea? Thanks

 

[azntoon]

.The form of the propagator gives us information about how the system evolves from one state to another over time. The exp[if] factor is a phase factor, so it does not change the magnitude of the amplitude, but just the phase. However, the term in the square root does depend on the difference between x'' and x', so the magnitude of the amplitude will change based on the difference in positions. This gives us an idea of how the system is evolving from one position to another over time.