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thermodynamic simulations & feynman quantum path integration 411Here T is the kinetic energy, V is the potential energy, ẋ = dx/dt, and square bracketsindicate a functional 3 of the function x(t) and ẋ(t).Feynman observed that the classical action for a free particle (V =0),S[b, a]= m 2 (ẋ)2 (t b − t a )= m 2(x b − x a ) 2t b − t a, (15.43)is related to the free-particle propagator (15.40) by√mG(b, a)=eiS[b,a]/¯h . (15.44)2πi(t b − t a )This is the much sought connection between quantum mechanics and Hamilton’sprinciple. Feynman then postulated a reformulation of quantum mechanics thatincorporated its statistical aspects by expressing G(b, a) as the weighted sum overall paths connecting a to b,G(b, a)= ∑e iS[b,a]/¯h (path integral). (15.45)pathsHere the classical action S (15.42) is evaluated along different paths (Figure 15.7),and the exponential of the action is summed over paths. The sum (15.45) is calleda path integral because it sums over actions S[b, a], each of which is an integral (onthe computer an integral and a sum are the same anyway). The essential connectionbetween classical and quantum mechanics is the realization that in units of¯h ≃ 10 −34 Js, the action is a very large number, S/¯h ≥ 10 20 , and so even though allpaths enter into the sum (15.45), the main contributions come from those pathsadjacent to the classical trajectory ¯x. In fact, because S is an extremum for the classicaltrajectory, it is a constant to first order in the variation of paths, and so nearbypaths have phases that vary smoothly and relatively slowly. In contrast, pathsfar from the classical trajectory are weighted by a rapidly oscillating exp(iS/¯h),and when many are included, they tend to cancel each other out. In the classicallimit, ¯h → 0, only the single classical trajectory contributes and (15.45) becomesHamilton’s principle of least action! In Figure 15.8 we show an example of atrajectory used in path-integral calculations.3 A functional is a number whose value depends on the complete behavior of some functionand not just on its behavior at one point. For example, the derivative f ′ (x) depends onthe value of f at x, yet the integral I[f]= ∫ bdx f(x) depends on the entire function andais therefore a functional of f.−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 411