From Path Integration to the Schrödinger Equation
We show how from the path integral we rederive the Schrodinger equation. This will confirm the Feynman path integration to be the formulation of quantum mechanics presenting an alternative way of examining the subject matter than the usual approaches with applications as vast as those of quantum mechanics itself.
Wave Function
We show how the wave function 'Pfij, t) = (q'P(f)) is the total amplitude for the particle to be found at (q,t), arriving from the past in some situation relates the transition amplitude K{qb,tb;q„,ta) describing a particle coming from and arriving at (qb,tb). The transition amplitude 4.98 relates {qb 'PCfj,))
when tc the intermediate time falls between the times ta and f(>:
Since the states have delta-normalization, then
so, the transition amplitude K(qb, tb,q,v ta) plays the role of the wave function when the initial condition is rigorously defined and this wave function must obey the following integral equation:
In fact, equation 7.3 is an integral equation of quantum mechanics, which is equivalent to the following time-dependent Schrodinger differential equation that we show later:
So, the transition amplitude К is the evolution operator in the representation of functional integrals and is also the wave function that has all the history of incident of the particle on the point qb. Usually, the wave function in ordinary quantum mechanics gives the probability that we find the particle at a given point and does not give the answer on where initially was found the particle. However, К gives the information on the point from where the particle left before being incident on the point qb. This implies К is the wave function where correspondingly is rigorously defined the initial state.
If the action functional in 4.82 is selected to be equal to the classical action 4.83 and equation 7.3 becomes the integral representation of the Schrodinger equation. The path-integral formulation of quantum mechanics fully is equivalent to the Schrodinger and Heisenberg pictures. When |<5S| is large compared to ft, i.e., when small changes in the path induce large changes in the action, then the phase factor
exp j changes rapidly and nearby paths will cancel each others contributions to the path integral
due to destructive interference. So, for paths with the action more than ft different from the classical path, interference becomes more and more destructive and there is lesser net contribution. Paths where the action is stationary (SS = 0) will experience the least destructive interference, so the largest contribution to the path integral is given by these classical paths. This implies paths close to the classical path have nearly the same action and thus the same phase and so will interfere constructively and make the dominating contributions to the transition amplitude. So, we expect paths whose action is within about ft of the classical path will contribute strongly. This links the path-integral formalism with the principle of stationary action in classical mechanics. This presents an elegant manner of thinking about quantum dynamics, and, at the same time, illustrating how the Heisenberg uncertainty limit comes about via wave (or path) interference.
