How Important is Spatial Invariance Symmetry to Quantum Mechanics? Part II

 In Part I, we argued that if spatial translational invariance is present physically in a problem then it should appear explicitly in equations describing the situation. In particular, if an equation exists with x absent, this suggests that it is valid for any x. We suggest, however, that one should try to seek equations consistent with the x-absent equation, which show x explicitly. 

 In Part I, we examined the case of photon reflection, and one dimensional photon reflection-refraction. We argued that such a problem might be described in terms of probabilities which are necessarily positive, i.e. AA/velocity, where AA is flux. No x is present, but one may write AA as A*A/velocity and introduce exp(i f(x,p)). Given that there exist exact x interaction points in the two problems and that the basic physics does not change if this x position changes, we suggested using continuity of exp(i f(x,p)) and its first derivative at the x-interaction point. These two equations combined yielded a pressure balance equation which does not depend on x, suggesting that exp(i f(x,p)) = ex(ipx), hence yielding the free particle wavefunction of quantum mechanics.

   In this note, we try to consider a free particle in a somewhat different view. We do not consider any reaction at first, such as reflection or reflection-refraction. Rather, we note that the particle with constant velocity has equal probability to be in any dx region. Thus, P(x)= dx/L, where L is length. P(x)>0, so it may be written as AA. There is x invariance in this problem and so we suggest there should be a corresponding equation(s) which yield AA, but explicitly show x. We suggest again exp(i f(p,x)). 

   At this point, there is no interaction so there is no continuity of exp(i f(p,x)) and its derivative at the interaction point. (There is no interaction point.) Instead, consider finding information about f(p,x) using ideas of entropy. A particle with P(x)=dx/L has equal probability to be at any x point and this suggests maximum entropy. For example, this kind of spatial probability is also used in the description of a Maxwell-Boltzmann (MB) gas. We suggest that maximum entropy should also apply to ex(i f(p,x)) and try to develop arguments to argue that exp(i f(p,x))=exp(ipx).

   Finally, we consider the case of a particle in a potential V(x). At first it seems that V(x) does not contain spatial invariance, unlike a mirror or index of refraction-reflection junction. If, however, one considers a potential to actually be an average expression describing a set of impulse hits, then, like in an MB gas, a particle may move freely until it suffers an impulse from the potential, not another particle as in the MB gas. In such a case, one may have all p values and there is translational invariance for a free particle. Thus, exp(ipx) should hold, but this is a probability (because it is linked to the a maximum entropy solution) and so one should be able to write W(x)=Sum over p  a(p)exp(ipx) and use this together with conservation of energy to solve the V(x) problem. We consider this in more detail.