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410 chapter 15Bt bx bx aTimet aAPositionFigure 15.7 A collection of paths connecting the initial space-time point A to the final pointB. The solid line is the trajectory followed by a classical particle, while the dashed lines areadditional paths sampled by a quantum particle. A classical particle somehow “knows”ahead of time that travel along the classical trajectory minimizes the action S.the different paths. All paths are possible, but some are more likely than others.(When you realize that Schrödinger theory solves for wave functions and considerspaths a classical concept, you can appreciate how different it is from Feynman’sview.) The values for the probabilities of the paths derive from Hamilton’s classicalprinciple of least action:The most general motion of a physical particle moving along the classicaltrajectory ¯x(t) from time t a to t b is along a path such that the action S[¯x(t)] isan extremum:δS[¯x(t)] = S[¯x(t)+δx(t)] − S[¯x(t)]=0, (15.41)with the paths constrained to pass through the endpoints:δ(x a )=δ(x b )=0.This formulation of classical mechanics, which is based on the calculus of variations,is equivalent to Newton’s differential equations if the action S is taken as the lineintegral of the Lagrangian along the path:∫ tbS[¯x(t)] = dt L [x(t), ẋ(t)] , L= T [x, ẋ] − V [x]. (15.42)t a−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 410