Comparison of Three Methods for Quadratic Potential Propagators Part III

Addendum March 15, 2021: In a new note, Part IV, we resolve the issue between using Q(t)exp(i Action) and postulating a form Q(t)exp{-i{b(t)XX + g(t) YY + g1(t) XY)} and inserting in a time-dependent Schrodinger equation for the case of an f(t)x dx/dt term in the Lagrangian. The two approaches are the same if one ensures that H=pdx/dt - L, as an extra f(t)f(t)xx/2 term appears in the Schrodinger equation, resolving the problem shown in this note. In a previous note (Part II), we considered three different approaches to calculating the propagator for the Schrodinger equation: id/dt W = -1/2m d/dx dW/dx + k(t)xx + f(t)x(-i)d/dx W ((1)). The first postulated the form Q(t)exp{-i [b(t)XX + g(t)YY + g2(t)XY]}, substituted it into the Schrodinger equation and equates coefficientd of XX, YY, XY separately (1). The other two methods (2),(3) linked the propagator to a classical Lagrangian. Method (2) used only classical solutions of the Lagrangian, while (3) postulated the form Q(t)exp(-i Action) for the propagator. (These approaches are applied to Lagrangians which are quadratic in X.) It was noted in (2), that f(t) x dx/dt may be written classically as: d/dt [.5xx f(t)] - .5xx df/dt. The term d/dt [.5xx f(t)] contributes nothing to the classical equations of motion and so is dropped. Method (3) does not drop such a term as the propagator is exp(-iAction) leading to an additional phase exp(-i Integral dt d/dt [.5xx f(t)]). Furthermore, even with this extra phase, exp(-i Action) does not satisfy the time dependent Schrodginer equation. In this note, we wish to examine why there seem to be issues with removing a d/dt[] piece from a Lagrangian when calculating a quantum propagator. We argue that a propagator linked to a classical Lagrangian i.e. exp(-iAction) for a free particle and a spring with k(t) as the spring constant leads to a coefficient of c dA/dt / A where A is a solution of the classical equations of motion. This behaviour is consistent with a fluctuation equation linking d/dt with d/dx d/dx and V(x, dx/dt) the potential i.e. the Schrodinger equation. Thus, we argue that the Schrodinger equation propagator is linked to classical equations of motion, but only for certain cases which produce a special kind of fluctuation. We argue that the extra phase exp(-i Integral dt d/dt (f(t).5xx)) does not satisfy this fluctuation rule. To clarify, we consider the free particle case, add a potential d/dt [.5xx f(t)]) and argue that the phase exp(-i Integral dt d/dt [.5xx f(t)]) multiplied by exp(ipx - iEt) is a solution only if x(t) is taken to be classical within the phase integral. This is not a quantum mechanical solution as it only considers classical motion and leads to a kind of hybrid equation mixing quantum fluctuations with a classical phase. Thus, it seems one must be careful when linking classical physics (i.e. the Lagrangian) to the Schrodinger equation