Space Time Slices in Quantum and Classical Mechanics Part II

Note: The equations  d/dx partial [ T L] = p and d/dt partial [ TL] = -E hold for both the relativistic and nonrelativistic free particles because dL/dv = p and H=pv-L hold for both the relativistic and nonrelativistic free particle.

In part I of this note, we argued that in quantum bound states one has a notion of dx and dt associated with p and E i.e. dx→1/p and dt→1/E, but not of continuous time associated with particle motion. exp(-iE t) contains time t, but this refers to a cycling time associated with  various exp(ip1x) free particle probabilities combining with exp(ikx) to form exp(ipx). In other words there is a certain time resolution as well as a space resolution (wavelength) for the single particle and time t refers to the entire quantum resonance. In a bound system, there is also a vrms(x) which holds for every quantum energy state En and this is linked to a continuous space time picture i.e. x(t). It is this x(t) which gives a sense of past, present and future. A quantum system, however, is a statistical system much like V(x) a potential. It does not follow a particle from one x to another in space because the potential renders stochastic hits. It is only interested in the overall resonance which forms. “t” in exp(-iEt) is associated with this resonance and not with single particle motion. Thus there is no classical sense of past, present or future in the quantum bound state at least with respect to the single particle. An entire quantum system may, however, evolve, but a bound system simply cycles through time. The single particle has no time attributes except dt associated with 1/E where E=pp/2m.

       In part 1 we argued that p may be thought of in terms of impulse (hits). In such a case it is not directly linked to space and time. We also argued that p=mo v even in the quantum case so dx and dt must be present indirectly. In this note we revisit these ideas by following (1) and deriving quantum mechanical behaviour for a free particle from the classical Lagarangian equations dL/dx (partial) = p and H= pv - L. This directly shows that a quantum picture deals with dx and dt for a single free particle and so the idea of x(t) only holds in an rms sense. For a single free particle, matters differ from the bound case because p = mo vrms (free particle). Thus we argue that exp(ipx) must be viewed in terms of probability and is only manifest if an interaction occurs.Such interactions, however, may be continuously occurring as in the case of the Bohm Aharonov effect in which there is a magnetic vector  potential existing in a large region of space.