Feynman Path Integrals in Quantum Mechanics

The diffusion equation in an external field (3.21) or (3.43) is the same as the nonrelativisllc Schrodlnger equation for a particle in an external field, apart from the appearance of the imaginary unit i which appears in the Schrodlnger equation. Specifically, this Schrbdinger equation for the 

1) we impose the boundary condition A = 0 if t /. Because of the similarity between (4.1) and (3.43), it is not surprising that K can be expressed as a functional integral. We could, at this point, quote the result, and then a derivation analogous to (3.26) to (3.43) would prove that indeed the result is the solution to (4.1). It is, however, instructive to derive the result as opposed to Feynman s approach of introducing it as a postulate, 

A numerical analyst might solve (4.1) by breaking the time interval 

Analogous to (3.6), the points r. are the discrete representations of the continuous curve r(/). Here r(t) is a possible trajectory followed by the particle. Thus the product of the A sin (4.3) gives the probability amplitude density that the particle follow the trajectory fj. As the integration is over all possible intermediate points r / = 1. n — 1, the multiple 

The theoretician tells us that the solution (4.3) is an exact solution to (4.1) provided that the e, are taken to be arbitrarily small, or according to 

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