exp(i Action) with V(x) and the Schrodinger Equation

The form exp(i Action) is often used as a propagator of solutions of the Schrdodinger equation. In (1), x(t)= Y/A(T) A(T-t) + X/A(T) A(t) ((1)) and this form is used in the Lagrangian to calculate Action = Integral dt1 (0,T) L. (Here we have interchanged X and Y from the form in ((1)).) We argue that unless X=A(T), this is not really a Lagrangian. In fact, X and A(T) are treated as independent in order to have exp(i Action) satisfy the Schrodinger equation. In an earlier note (2), we examined the free particle case for which Integral (0,T) dt1 mvv/2 = T X/T X/T m/2. exp(i .5m XX/T) has certain properties under the operations of d/dT and d/dX. In particular, id/dT exp(i Action) = E exp(i Action) and -id/dX exp(i Action) = p exp(i Action). These values may be inserted into an energy conservation equation containing E and p and lead to a differential equation i.e. the Schrodinger equation (or Klein-Gordon equation in the relativistic case). We argued in (2) that one may generalize the free particle Schrodinger equation by adding a potential V(x) which is considered to be stochastic i.e. V(x)= Sum over k Vk exp(ikx). In such a case, one has a momentum distribution P(p/x)= a(p) exp(i px) / W(x) where W(x)= Sum over p a(p) exp(ipx) instead of the free particle exp(ipx). In this note, we consider extending exp(i .5m XX/T), the nonrelativistic exp(i Action) to a full “Lagrangian”. We point out that this is not really the Lagrangian because X and A(T) are considered to be independent which they are not. (A(t) is a solution of Newton’s law.) Thus, we seek an object such that id/dT object + V(x) = -1/2m d/dXobject * d/dXobject + V(x). Here V(x) is the potential which was added to the free particle Schrodinger equation as part of conservation of energy. We argue that exp( i Action) where Action uses the full Lagrangian with the “form” ((1)) satisfies the Schrodinger equation