Particle-Wave Duality and Two Types of Measurements

We argue that in classical mechanics there exist at least two types of measurements of a constant moving particle. (This may be extended to an accelerating one.) First there is the x(t) description which is sometimes considered a complete description of the mechanical case. This implies one may measure position and time of the particle successively and thus know about its behaviour in space and time. We argue that there is a second measurement which is not one of space or time, but rather of impulse. A constant moving particle may strike a foil at any place in its path at any time (if one has an ensemble of such classical particles) and the foil will “recognize” that it has been hit by a momentum p.

    In quantum mechanics we argue that there are also two types of measurement for a free particle. The first is the identical momentum impulse measurement i.e. the particle strikes a foil which recognizes it has momentum p. There seems to be no quantum feature to this. Such a measurement has already been performed classically for photons. This is the “particle” like measurement which has nothing to do with wave features. A. Einstein in 1905 in his work on the photoelectric effect already noted that particle type behaviour of a photon. The second type of measurement deals with spatial features which are very different from x(t). A quantum free particle is represented by exp(ipx) which contains “p” which may be measured by an impulse. We argue, however, that spatial features, unlike those for a classical wave in which the medium and wave disturbance are separated, require interference i.e.  the summing of exp(ipx) and exp(i p1 x) or exp(i p1 x) or exp(i p (x+b)) or exp(i p1 (x+b1)). Thus exp(ipx) behaves like a complex probability. Mathematically it has a modulus of 1 which is said to be linked to equal probability for it to be at any x. Why should this be relevant? We argue that it is because of the impulse measurement which indicates that a p hit may occur at any x. We further argue that exp(ipx) behaves as part of a projection operator because it is orthogonal to other exp(ip1x)’s.

It is this orthogonality which seems to allow for one to build a probability medium, but requires W*(x)W(x)=P(x). In such a case, a*(p)a(p) terms create the medium from which probability may be added and removed by cross terms exp(ip1 x) exp(ip2 x). Orthogonality ensures that overall what is added and removed cancel. Thus there are two levels of probability, one at W(x) linked to OR situations created by interactions which reveal the spatial nature of a quantum free particle and the other W*(x)W(x).