Application to Schrödinger Pseudo-Hamiltonians

Abstract

In the final chapter, we are going to use the path-integral methods discussed above to treat the dissipative systems whose time evolution is governed by Schrödinger pseudo-Hamiltonians. First we prove the formula that expresses the propagator of such a system by means of an appropriate F-integral for various classes of absorptive complex potentials, including the time-dependent ones. As an illustration, we present in Section 2 a detailed discussion of a multidimensional damped harmonic oscillator described by a complex quadratic potential; its propagator is obtained explicitly by evaluating the respective product F-integral. Finally, in Section 3 we return to the argument which is employed usually to motivate the path-integral expression of time-evolution operators. We show that it is also applicable to dissipative systems, though they may have no classical counterpart.

”In part, the point of functional integration is a less cumbersome notation, but there is a larger point: like any other successful language, its existence tends to lead us to a different and very special way of thinking.”

B. Simon