Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1007corresponding Hermitian operators in the Hilbert space, with real measurableeigenvalues, which obey Heisenberg commutation relations.The path–integral quantization is instead based directly on the notion ofa propagator K(q f , t f ; q i , t i ) which is defined such that (see [Ryder (1996);Cheng and Li (1984); Gunion (2003)])∫ψ(q f , t f ) = K(q f , t f ; q i , t i ) ψ(q i , t i ) dq i , (6.12)i.e., the wave function ψ(q f , t f ) at final time t f is given by a Huygensprinciple in terms of the wave function ψ(q i , t i ) at an initial time t i , wherewe have to integrate over all the points q i since all can, in principle, sendout little wavelets that would influence the value of the wave function atq f at the later time t f . This equation is very general and is an expressionof causality. We use the normal units with = 1.According to the usual interpretation of quantum mechanics, ψ(q f , t f )is the probability amplitude that the particle is at the point q f and thetime t f , which means that K(q f , t f ; q i , t i ) is the probability amplitude fora transition from q i and t i to q f and t f . The probability that the particleis observed at q f at time t f if it began at q i at time t i isP (q f , t f ; q i , t i ) = |K(q f , t f ; q i , t i )| 2 .Let us now divide the time interval between t i and t f into two, with tas the intermediate time, and q the intermediate point in space. Repeatedapplication of (6.12) gives∫ ∫ψ(q f , t f ) = K(q f , t f ; q, t) dq K(q, t; q i , t i ) ψ(q i , t i ) dq i ,from which it follows that∫K(q f , t f ; q i , t i ) =dq K(q f , t f ; q, t) K(q, t; q i , t i ).This equation says that the transition from (q i , t i ) to (q f , t f ) may be regardedas the result of the transition from (q i , t i ) to all available intermediatepoints q followed by a transition from (q, t) to (q f , t f ). This notionof all possible paths is crucial in the path–integral formulation of quantummechanics.Now, recall that the state vector |ψ, t〉 Sin the Schrödinger picture isrelated to that in the Heisenberg picture |ψ〉 Hby|ψ, t〉 S= e −iHt |ψ〉 H,