Collective and Stochastic Motion in the Time-Dependent Schrodinger Equation Part 3: Propagators

In a previous note (1) Part 2, we compared two approaches to solving the Schrodinger equation: id/dt (partial) W(x,t) = -1/2m d/dx d/dx W + .5kxx W + f(t)x W ((1)), with both utilizing the idea of a transformation, namely y=x-b(t). The first approach involved unitary transformations, while the second changing variables and searching for a product wavefunction WoW1 such that Wo satisfies the oscillator equation without f(t)x (but with a c(t)W term easily handled by a time phase) and W1 satisfies a Schrodinger like equation containing f(t)y and a coupling term -1/2m dWo/dy dW1/dy such that i d/dt partial W1 cancels this term. In this note, we consider a third approach, that of propagators outlined in (2). This method may be used to find the propagator of ((1)). Interestingly, following (2), one may postulate a propagator solution of the form Go(x,t)exp[i ( a(t)x + b(t)x1 + g(t))] where Go(x,x1,t) is the oscillator propagator (without xf(t)). Thus, one does not ever use the transformation y=x-b(t) unlike the previous two approaches. This different approach, however, requires explicit knowledge of Go(x,x1,t), the oscillator propagator. There are no “cancellations”. Given that the two previous approaches involve a solution containing Wn(y), where Wn is the time-independent oscillator solution in the variable y=x-b(t), one might expect that a propagator for ((1)) should use a Go(y,t) and not Go(x,t). We show one may obtain the same results, by considering the use of a transformation y together with a propagator. Using y involves solving one differential equation as opposed to three for the non-y method. In addition, we consider applying the approach of (2) to the calculation of W(x,t). We find that exp(ia(t)x +ig(t)] must be modified to exp(ia(t)x + B(t)x + ig(t)] which may then be used to find the ground state solution of the oscillator plus xf(t) Hamiltionian utilizing the explicit form of Wo(x,t) the ground state. We argue that the transformation approach seems to be more convenient for higher energy level