Comparison of Three Methods for Quadratic Potential Propagators

In this note, we compare three methods for calculating a propagator for a problem with a potential of the form:b(t)xx + d(t)x dx/dt ((1)) and constant mass. Later, we consider making the mass time dependent as well. The first method is that of (1) for which the form Q(t)exp{ -i [R(t)XX + V(t)YY + Z(t)XY]} is postulated for the propagator and inserted into the time-dependent Schrodinger equation. Coefficients of various powers of X and Y are collected and taken to be zero as X and Y may be varied independently. This yields a set of differential equations for Q(t), R(t), V(t) and Z(t). A second method is outlined in (2) and computes the propagator for ((1)) directly using solutions of the classical Newton’s second law (without integrating them). This gives a structural form for the propagator, for example R(t) is of the form dB/dt / B where B(t) is a function depending linearly on a solution to Newton’s second law. It should be noted that this method suggests that a potential term in a classical Lagrangian b(t)x dx/dt may be rewritten as d/dt [g(t)xx] - g1(t)xx. Thus, the potential is written as .5c(t)xx with the d/dt [] piece dropped as it has no consequences on the classical solutions. (We find that this leads to different results from (1).) The third method is that of the classical action (3) i.e. calculating the propagator as Q(t)exp(i Sc) where Sc is the classical action which depends on solutions of Newton’s law which are inserted into the Lagrangian which is then integrated over time. In particular, we wish to show how the classical action approach is directly linked to the other two methods. The three methods are identical if the mass is constant and the quadratic potential is of the form .5c(t)xx with no b(t)x dx/dt piece. For methods 2 and 3 which use classical Lagrangians, a b(t)x dx/dt potential term may be written as: d/dt (g(t)xx) -g2(t)xx with the d/dt [g(t)xx) being dropped from the Lagrangian as it does not affect the equations of motion. For the time-dependent Schrodinger equation, one only has the Schrodinger equation with the fixed potential which cannot be changed. Thus, it seems that in the case of a b(t)x(-idW/dx) term, the propagator obtained by this method differs with that of (2) if the xx and x dx/dt potentials are converted into a strictly xx potential. If both potential pieces are left, then the method of (3) is still linked to (1), but in a more complicated manner. For the case of a pure .5c(t)xx potential in the Schrodinger equation, the coefficient of XX from method (1) is proportional to dA/dt / A, where A is a solution of Newton’s second law