Comparison of Three Methods for Quadratic Potential Propagators Part IV

  In a series of previous notes (i.e. Parts II and III) we considered three methods for computing the propagator for the problem:  id/dt W = -1/2m d/dx d/dx W + k(t)xxW + f(t)x (-id/dx) W. The first method is based on postulating a form Q(t) exp(-i{ b(t)XX + g(t)YY +g2(t)XY}), inserting into the time-dependent Schrodinger equation and equating coefficients of XX etc The other two methods are based on classical Lagrangians. We argued in Parts II and III that for f(t)=0 there is agreement between the methods, but issues arise with the presence of f(t)x (-id/dx) W. In this note, we try to resolve this issue by using H= p dx/dt - L ((1)) where L is the Lagrangian. 

    In method 2, a Lagrangian with f(t)x dx/dt is considered, but this term is converted to d/dt (.5xx f) - .5xx df/dt. Thus, the time-dependent oscillator potential is modified and the d/dt [] piece thrown away because it does not affect the classical equations of motion. Method 3 (3) computes the propagator using Q(t)exp(i Action), so d/dt [] is included and leads to a phase.

By using ((1)) we argue that if f(t)x dx/dt exists in the Lagrangian, one must consider a corresponding Hamiltonian of the form:  H= [p+f(t)x)][p+f(t)x)] (1/2m) + k(t)xx. This is equivalent to:   pp/2m + f(t)f(t)xx (1/2m) + k(t)xx + f(t)(1/m)xp -i f/2m.  Thus, we see an extra f(t)f(t)xx/2m in the Hamiltonian in addition to the f(t)xp type term. This “fixes” the problem of “”ff”” terms not canceling from the Schrodinger equation in parts II and III. Thus, one must be careful when dealing with velocity dependent potential, but also note that p=mdx/dt -f(t)x now so p as in the electromagnetic case is not simply the momentum of the particle.