Quantum Wavefunction and Fluctuation of Classical Action Part III

 In Part I of this note, we argued that d/dX Action (where Action = Integral dt L, L being the Lagrangian) for a particle with constant velocity is p whether the classical L=.5mvv or the relativistic L=-mo sqrt(1-vv) (c=1) is used. Thus, a fluctuation or error of delta X leads to a change of (delta X) p in the Action. We tried to argue that this led to the form exp(ipx).

    In this note, we revisit this scenario. We argue that if one wishes to have a constant error due to a fluctuation/error in X in the Action regardless of p, then delta x = 1/p is a characteristic length. At first one may expect a relative error to be of interest, but we argue that a potential may knock a particle from one p value to another. In such a case, it may be useful to have the error, and not the relative error, remain constant. Thus, one already has a wavelength, but this suggests a periodic function to describe the error of the action throughout space, and there are many possibilities. A periodic function may have positive and negative values corresponding to positive and negative X fluctuations in a regular way. We ultimately argue that exp(ipx) is ideal and is also linked to spatial density through exp(-ipx) exp(ipx)=1. Thus, we argue that two ideas are involved. First, the idea of an error/fluctuation in the Action is used to develop the idea of a wavelength. Secondly, one tries to find a periodic function associated with the wavelength which is also linked to spatial density. Finally, one tries to generalize these ideas to the nonrelativistic case of potential present. This seems to present a statistical picture.